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\usepackage{numerica}

\newcommand\rel{\,\varrho\;}
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\begin{document}
\title{\texttt{numerica-basics}}
\author{Andrew Parsloe\\
(\url{ajparsloe@gmail.com})\\
}
\maketitle
\begin{abstract}
The \verb`numerica` package defines a command to wrap around a mathematical
expression in its LaTeX form and, once values are assigned to variables,
numerically evaluate it. The intent is to avoid the need to modify
the LaTeX form of the expression being evaluated. For programs with
a preview facility like LyX, or compile-as-you-go systems, interactive
back-of-envelope calculations and numerical exploration are possible
within the document being worked on. The package requires the bundles
\verb`l3kernel` and \verb`l3packages`, and the \verb`amsmath` and
\verb`mathtools` packages. Additional modules define commands to
iterate and find fixed points of functions of a single variable, to
find the zeros or extrema of such functions, to calculate the terms
of recurrence relations, and to create multi-column tables of function
values (which requires the \verb`booktabs` package).\\
\\
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\paragraph*{Note:}
\begin{itemize}
\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica.sty}}{\normalsize .}{\small\par}
\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
\item {\normalsize The package requires }{\normalsize\texttt{amsmath}}{\normalsize{}
and }{\normalsize\texttt{mathtools}}{\normalsize .}{\small\par}
\item {\normalsize I refer many times in this document (especially §\ref{sec:Argument-parsing})
to }{\normalsize\emph{Handbook of Mathematical Functions}}{\normalsize ,
edited by Milton Abramowitz and Irene A. Segun, Dover, 1965. This
is abbreviated to }{\normalsize\emph{HMF}}{\normalsize , often followed
by a number like 1.2.3 to locate the actual expression referenced.}{\small\par}
\end{itemize}
\end{shaded}%
\end{minipage}
\end{abstract}
\begin{center}
\tableofcontents{}
\par\end{center}

\chapter{Introduction}

\texttt{numerica} is a \LaTeX{} package offering the ability to numerically
evaluate mathematical expressions in the \LaTeX{} form in which they
are typeset.\footnote{\texttt{numerica} evolved from the author's \texttt{calculyx} package
that was designed for use with the document processor \LyX{} and available
for download from a link on the \LyX{} wiki website (but not from CTAN). }

There are a number of packages which can do calculations in \LaTeX ,\footnote{A simple search finds the venerable \texttt{calc} in the \LaTeX{} base,
\texttt{calculator }(including an associated \texttt{calculus} package),
\texttt{fltpoint}, \texttt{fp} (\emph{fixed} rather than floating
point), \texttt{spreadtab} (using either \texttt{fp} or \texttt{l3fp}
as its calculational engine) if you want simple spreadsheeting with
your calculations, the elaborate \texttt{xint}, \texttt{pst-calculate}
(a limited interface to \texttt{l3fp}), \texttt{l3fp} in the \LaTeX 3
kernel, and \texttt{xfp}, the \LaTeX 3 interface to \texttt{l3fp}.
Other packages include a calculational element but are restricted
in their scope. (\texttt{longdivision} for instance is elegant, but
limited only to long division.) } but those I am aware of all require the mathematical expressions
they operate on to be changed to an appropriate syntax. Of these packages
\texttt{xfp} comes closest to my objective with \texttt{numerica}.
For instance, given a formula
\begin{centred}
\verb`\frac{\sin (3.5)}{2} + 2\cdot 10^{-3}`
\end{centred}
(in a math environment), this can be evaluated using \texttt{xfp}
by transforming the expression to \verb`sin(3.5)/2 + 2e-3` and wrapping
this in the command \verb`\fpeval`. In \texttt{numerica} you don't
need to transform the formula, just wrap it in an \verb`\eval` command
(for the acutal calculation see §\ref{subsec:introSimple-examples}):
\begin{centred}
\verb`\eval{ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3} }`.
\end{centred}
\texttt{numerica}, like \texttt{xfp} and a number of other packages,
uses \texttt{l3fp} (the \LaTeX 3 floating point module in \texttt{l3kernel})
as its calculational engine. To some extent the main command, \verb`\nmcEvaluate`,
short-name form \verb`\eval`, is a pre-processor to \texttt{l3fp},
converting mathematical expressions written in the \LaTeX\texttt{
}form in which they will be typeset into an `fp-ified' form that
is digestible by \texttt{l3fp}. The aim is to make the command act
as a wrapper around such formulas. Ideally, one should not have to
make \emph{any} adjustment to them, although any text on Fourier series
suggests that hope in full generality is delusional. Surprisingly
often however it \emph{is} possible. We will see shortly that even
complicated formulas like

\[
\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
\]
and
\[
\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
\]

\noindent can be evaluated `as is' (see below, §\ref{subsec:introReassurance}).
There is no need to shift the position of the superscript $2$ on
the sines, no need to parenthesize the arguments of $\sin$ and $\cos$,
no need to insert asterisks to indicate multiplication, no need to
change the \verb`\frac` and \verb`\tfrac`-s to slashes, \texttt{/},
no need to delete the \verb`\left` and \verb`\right` that qualify
the big parentheses (in the underlying \LaTeX ) in the second expression.
Of course, if there are variables in an expression, as in these examples,
they will need to be assigned values. And how the result of the evaluation
is presented also requires specifying, but the aim is always: to evaluate
mathematical expressions in \LaTeX{} with as little adjustment as possible
to the form in which they are typeset.

\texttt{numerica} is written in \texttt{expl3}, the programming language
of the \LaTeX 3 project. It uses the \LaTeX 3 module \texttt{l3fp}
(part of \texttt{l3kernel}) as its calculational engine. This enables
floating point operations to 16 significant figures, with exponents
ranging between $-10000$ and $+10000$. Many functions and operations
are built-in to \texttt{l3fp} – arithmetic operations, trigonometric,
exponential and logarithm functions, factorials, absolute value, max
and min. Others have been constructed for \texttt{numerica }from \texttt{l3fp}
ingredients – binomial coefficients, hyperbolic functions, sums and
products – but to the user there should be no discernible difference. 

Associated modules provide for additional operations: iteration, finding
zeros, recurrence relations, mathematical table building. Further
modules are planned (e.g. calculus).

\section{How to use \texttt{numerica}}

The package is invoked in the usual way:\footnote{I use the angle-bracket notation to indicate optional user input.
Of course what is input does not include the angle brackets.} put
\begin{lyxcode}
\textbackslash usepackage{[}<options>{]}\{numerica\}
\end{lyxcode}
\noindent in the \LaTeX{} preamble. \texttt{numerica} requires the
\texttt{amsmath} and \texttt{mathtools} packages and loads these automatically.
\texttt{numerica} will also accept use of some relational symbols
from the \texttt{amssymb} package provided that package is loaded;
see §\ref{subsec:evalBoolean-output}. 

\subsection{Packages and package options}

\label{subsec:introPackagesOptions}Version 1.0.0 of \texttt{numerica}
has three package options. 
\begin{description}
\item [{\texttt{plus}}] By calling \texttt{numerica} with the \texttt{plus}
package option,\\
{\ttfamily{}%
\noindent\begin{minipage}[t]{1\columnwidth}%
~~~~\textbackslash usepackage{[}plus{]}\{numerica\}%
\end{minipage}}\texttt{}~\\
the file \texttt{numerica-plus.def }is loaded where a number of additional
commands: \verb`\nmcIterate`, \verb`\nmcSolve`, \verb`\nmcRecur`
are defined. These enable the iteration of functions of a single variable\footnote{At this stage!},
including finding fixed points; the solving of equations of the form
$f(x)=0$ (or the location of local maxima or minima); and the calculation
of terms in recurrence relations (like the Fibonacci series or othogonal
polynomials). See the associated document \texttt{numerica-plus.pdf}.
\item [{\texttt{tables}}] By calling \texttt{numerica} with the \texttt{tables}
package option\\
{\ttfamily{}%
\noindent\begin{minipage}[t]{1\columnwidth}%
~~~~\textbackslash usepackage{[}tables{]}\{numerica\}%
\end{minipage}}\texttt{}~\\
the file \texttt{numerica-tables.def} is loaded with with the command
\verb`\nmcTabulate` enabling the creation of multi-column tables
of function values with a wide variety of formatting options (most
of those employed in \emph{HMF} in fact). See the associated document
\texttt{numerica-tables.pdf}.
\item [{\texttt{lyx}}] By calling \texttt{numerica} with the \texttt{lyx}
package option,\\
{\ttfamily{}%
\noindent\begin{minipage}[t]{1\columnwidth}%
~~~~\textbackslash usepackage{[}lyx{]}\{numerica\}%
\end{minipage}}\texttt{}~\\
the file \texttt{numerica-lyx.def }is loaded with code enabling the
full use of the \verb`\nmcReuse` command in the document processor
\LyX{} (along with all other commands of the \texttt{numerica} package).
Use of \texttt{numerica} in \LyX{} exploits the mini-\LaTeX{} runs of
the instant preview facility of that program to give immediate feedback
on calculations without requiring the whole document to be compiled.
See Chapter~\ref{chap:LyX}.
\end{description}
More than one option can be used at a time by separating the options
with a comma; e.g. \texttt{\textbackslash usepackage{[}plus,tables{]}\{numerica\}}.
However, apart from Chapter~\ref{chap:LyX}, the present document
focuses on \texttt{numerica} when called with no options: \texttt{\textbackslash usepackage\{numerica\}}.

\subsection{Simple examples of use}

\label{subsec:introSimple-examples}A simple example of use is provided
by the document
\begin{verbatim}
    \documentclass{minimal} 
    \usepackage{numerica}
    \begin{document}

        \eval{$ mc^2 $}[m=70,c=299 792 458][8x]

    \end{document}

\end{verbatim}
\noindent We have a formula between math delimiters: \verb`$ mc^2 $`.
We have wrapped a command \verb`\eval` around the lot, added an optional
argument in parentheses specifying numericaal values for the quantities
\texttt{m} and \texttt{c}, and concluded it all with a trailing optional
argument specifying that the result should be presented to $8$ places
of decimals and in scientific notation (the \verb`x`). Running \texttt{pdflatex}
on this document generates a pdf displaying
\begin{centred}
\eval{$ mc^2 $}[m=70,c=299 792 458][8x]
\end{centred}
\noindent where the formula ($mc^{2})$ is equated to the numerical
value resulting from substituting the given values of $m$ and $c$.
Those values are displayed in a list following the result. The calculation
is presented to $8$ decimal places in scientific notation. (According
to Einstein's famous equation $E=mc^{2}$ this is the enormous energy
content, in joules, of what was once considered an average adult Caucasian
male. Only a minute fraction is ever available.) 

A second example is provided by the formula in earlier remarks:
\begin{verbatim}
    \documentclass{minimal} 
    \usepackage{numerica}
    \begin{document}

        \eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}

    \end{document}

\end{verbatim}
Running \texttt{pdflatex} on this document produces the result

\eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}

The \verb`\eval` command used in these examples is the main command
of the \texttt{numerica} package. It is discussed in full in the next
chapter, but first some preliminaries.

\subsection{Display of the result}

\label{subsec:introResultDisplay}In what follows I shall write things
like (but generally more complicated than)
\begin{centred}
\verb`$ \eval{ 1+1 } $` $\Longrightarrow \eval{ 1 + 1 } $
\end{centred}
to mean: run \texttt{pdflatex} on a document containing \verb`\eval{1+1}`
in the document body to generate a pdf containing the calculated result
($2$ in this instance). In this case the \verb`\eval` command is
used \emph{within} a math environment (delimited by the dollar signs).
It is not limited to this behaviour. The command can also wrap \emph{around}
the math delimiters (as we saw in the previous examples):
\begin{centred}
\verb`\eval{$ 1+1 $}` $\Longrightarrow$ \eval{$ 1+1 $}.
\end{centred}
As you can see, the display that results is different. 
\begin{itemize}
\item When the \verb`\eval` command is used\emph{ within} a math environment,
only the \emph{result,} followed possibly by the \emph{variable~=~value
list} (see §\ref{subsec:evalVv-list}) is displayed.
\end{itemize}
Environments may include the various AMS environments as well as the
standard \LaTeX{} inline ( \verb`$ $` ), \texttt{equation} ( \verb`\[ \]`
) and \texttt{eqnarray} environments. For an example of \verb`\eval`
within an \verb`align*` environment see §\ref{subsec:introExploring}
below. 
\begin{itemize}
\item When the \verb`\eval` command is wrapped\emph{ around} a math environment,
the result is displayed in the form, \emph{formula = result} (followed
possibly by the \emph{variable~=~value list}) within that environment,
\begin{itemize}
\item If the formula is long or contains many variables then it may be desirable
to split the display over two lines; see §\ref{subsec:evalChanging-display-format}
and §\ref{subsec:settings New-line-display},
\end{itemize}
\end{itemize}
the whole presented as an inline expression if \verb`$` delimiters
are used, or as a display-style expression otherwise. (See the $mc^{2}$
example for an illustration.) 

It is not clear to me that wrapping \verb`\eval` \emph{around} the
AMS environments, except for \texttt{multline}, makes much sense,
although it can be done. Here is an example of \verb`\eval` wrapped
around a \texttt{multline{*}} environment (the phantom is there so
that the hanging $+$ sign spaces correctly),\medskip{}

\begin{minipage}[t]{0.8\columnwidth}%
\begin{verbatim}
\eval{ \begin{multline*}
          1+2+3+4+5+6+7+8+9+10+\phantom{0}\\
            11+12+13+14+15+16+17+18+19 
       \end{multline*} }
\end{verbatim}
%
\end{minipage}

\medskip{}$\Longrightarrow$ %
\begin{minipage}[t]{0.8\columnwidth}%
\vspace{-2ex}\eval{ \begin{multline*}       
		1+2+3+4+5+6+7+8+9+10+\phantom{0}\\       		  11+12+13+14+15+16+17+18+19      
\end{multline*} }%
\end{minipage}
\begin{itemize}
\item It is also possible to dispense with math delimiters entirely, neither
wrapped within nor wrapped around the \verb`\eval` command, but in
that case \texttt{numerica} acts as if \verb`\eval` had been used
within \verb`\[` and \verb`\]` and displays the result accordingly.
\end{itemize}

\subsection{Exploring}

\label{subsec:introExploring}When working on \texttt{numerica}'s
predecessor package, I constantly tested it against known results
to check for coding errors. One test was to ensure that 
\[
\left(1+\frac{1}{n}\right)^{n}
\]
did indeed converge to the number $e$ as $n$ increased.\texttt{ }Let's
do that here. Try first $n=10$:
\begin{center}
\verb`\eval{$ e-(1+1/n)^n $}[n=10][x]` $\Longrightarrow$ \eval{$ e-(1+1/n)^n $}[n=10][x].
\par\end{center}

\noindent (The default number of decimal places displayed is $6$.)
The difference between $e$ and $(1+1/n)^{n}$ is about an eighth
($0.125$) when $n=10$, which is encouraging but hardly decisive.
The obvious thing to do is increase the value of $n$. I'll use an
\verb`align*` environment to `prettify' the presentation of the
results:
\begin{verbatim}
  \begin{align*}
    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^5][*x],\\
    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^6][*x],\\
    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^7][*x],\\
    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^8][*x].
  \end{align*}
\end{verbatim}
(most of which was written using copy and paste) which produces

\begin{align*}
e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{5}][x],\\
e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{6}][*x],\\
e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{7}][x],\\
e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{8}][x].
\end{align*}
Clearly $(1+1/n)^{n}$ converges to $e$, the difference between them
being of order $1/n$, but that is not what catches the eye. There
is an unanticipated regularity here. 1.35914? Double the number: \verb`$\eval{2\times 1.35914}[5]$`\texttt{
}$\Longrightarrow\eval{2\times1.35914}[5]$ which is close enough
to $e$ to suggest a relationship, namely,
\[
\lim_{n\to\infty}n\left(e-\left(1+\frac{1}{n}\right)^{n}\right)=\tfrac{1}{2}e.
\]

This was new to me. Is it true? From the familiar expansion of the
logarithm

\begin{align*}
\ln\left(1+\frac{1}{n}\right)^{n} & =n\ln\left(1+\frac{1}{n}\right)\\
 & =n\left(\frac{1}{n}-\frac{1}{2}\frac{1}{n^{2}}+\frac{1}{3}\frac{1}{n^{3}}-\ldots\right)\\
 & =1-\frac{1}{2n}\left(1-\frac{2}{3}\frac{1}{n}+\frac{2}{4}\frac{1}{n^{2}}-\right)\\
 & \equiv1-\frac{1}{2n}E_{n},
\end{align*}
say. Since $E_{n}$ is an alternating series and the magnitudes of
the terms of the series tend to $0$ monotonically, $1>E_{n}>1-2/3n$.
From this and the inequalities $1/(1-x)>e^{x}>1+x$ when $x<1$ it
proved a straightforward matter to verify the proposed limit.

\subsection{Reassurance}

\label{subsec:introReassurance}In the course of some hobbyist investigations
in plane hyperbolic geometry I derived the formula
\[
\Phi_{1}(m,n)=\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
\]
for $m=2,3,\ldots$ and integral $n\ge2m+1$. A key concern was: when
is $\Phi_{1}$ positive? After an embarrassingly laborious struggle,
I managed to work this expression into the form
\[
\Phi_{2}(m,n)=\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
\]
in which the conditions for positivity are clear: with $n\ge2m+1$,
so that $m\pi/3n<\pi/6$, the first factor is always positive, the
second is positive for $m\ge2$, and the third is positive for $m\ge4$.
All well and good, but given the struggle to derive $\Phi_{2}$, was
I confident that $\Phi_{1}$ and $\Phi_{2}$ really\emph{ }are equal?
It felt all too likely that I had made a mistake. 

The simplest way to check was to see if the two expressions gave the
same numericaal answers for a number of $m,\thinspace n$ values.
I wrote \verb`\eval{\[ \]}[m=2,n=5]` twice and between the delimiters
pasted the already composed expressions for $\Phi_{1}$ and $\Phi_{2}$,
namely: 
\begin{verbatim}
    \eval{\[ 
            \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
            \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
            {2\sin^{2}\tfrac{m}{3n}\pi} 
          \]}[m=2,n=5]
    \eval{\[ 
            \left(
              \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
              {2\sin^{2}\tfrac{m}{3n}\pi}
            \right)
            \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
          \]}[m=2,n=5]
\end{verbatim}
I have added some formatting – indenting, line breaks – to make the
formulas more readable for the present document but otherwise left
them unaltered. The \verb`\eval` command can be used for even quite
complicated expressions without needing to tinker with their \LaTeX{}
form, but you may wish – as here – to adjust white space to clarify
the component parts of the formula. Running \texttt{pdflatex} on these
expressions, the results were

\eval{\[ 
        \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
        \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
        {2\sin^{2}\tfrac{m}{3n}\pi} 
      \]}[m=2,n=5]

\eval{\[ 
        \left(
          \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
          {2\sin^{2}\tfrac{m}{3n}\pi}
        \right)
       \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
      \]}[m=2,n=5]

\noindent which was reassuring. Doing it again but with different
values of $m$ and $n$, again the results coincided:

\eval{\[ 
        \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
        \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
        {2\sin^{2}\tfrac{m}{3n}\pi} 
      \]}[m=5,n=13]

\eval{\[ 
        \left(
          \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
          {2\sin^{2}\tfrac{m}{3n}\pi}
        \right)
       \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
      \]}[m=5,n=13]

\noindent Thus reassured that there was \emph{not }an error in my
laborious derivation of $\Phi_{2}$ from $\Phi_{1}$, it was not difficult
to work back from $\Phi_{2}$ to $\Phi_{1}$ then reverse the argument
to find a straightforward derivation. 

\chapter{\texttt{\textbackslash nmcEvaluate} (\texttt{\textbackslash eval)}}

The main calculational command in \texttt{numerica} is \verb`\nmcEvaluate`.
Unlike some other commands which are loaded optionally, \verb`\nmcEvaluate`
is \emph{always} loaded, and therefore always available. Because \verb`\nmcEvaluate`
would be tiresome to write too frequently,\texttt{ }particularly for
back-of-envelope calculations, there is an equivalent short-name form,
\verb`\eval`, used almost exclusively in the following. But note:
wherever you see the command \verb`\eval`, you can substitute \verb`\nmcEvaluate`
and obtain the same result.

\verb`\eval` (like other short-name forms of other commands in the
\texttt{numerica} suite) is defined using \verb`\ProvideDocumentCommand`
from the \texttt{xparse} package. Hence if \verb`\eval` has already
been defined in some other package already loaded, it will not be
redefined by \texttt{numerica}. It will retain its meaning in the
other package. Its consequent absence from \texttt{numerica} may be
an irritant, but only that; \verb`\nmcEvaluate` is defined using
\texttt{xparse}'s \verb`\DeclareDocumentCommand` which would override
any (freakishly unlikely) previous definition of \verb`\nmcEvaluate`
in another package and would therefore still be available.

\section{Syntax of \texttt{\textbackslash nmcEvaluate (\textbackslash eval)}}

There are five arguments to the \verb`\nmcEvaluate` (or \verb`\eval`)
command, of which only one, the third, is mandatory. All others are
optional. If all are deployed the command looks like
\begin{centred}
\noindent \verb`\eval*[settings]{expr.}[vv-list][num. format]`
\end{centred}
I discuss the various arguments in the referenced sections.
\begin{enumerate}
\item \verb`*` optional switch; if present ensures display of only the
numerical result (suppresses display of the formula and vv-list);
see §\ref{subsec:evalVvSuppresList}
\item \verb`[settings]` optional comma-separated list of \emph{key=value
}settings for this particular calculation; see §\ref{sec:settingsOption}
\item \verb`{expr.}` the only mandatory argument; the mathematical expression/formula
in \LaTeX{} form that is to be evaluated 
\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
}items; see §\ref{subsec:evalVv-list}
\item \verb`[num. format]` optional format specification for presentation
of the numerical result (rounding, padding with zeros, scientific
notation, boolean output); see~§\ref{subsec:evalRoundingEtc}
\end{enumerate}
Note that arguments 4 and 5 are both square-bracket delimited optional
arguments. Should only one such argument be used, \texttt{numerica}
determines which is intended by looking for an equals sign within
the argument. Its presence indicates the argument is the vv-list;
its absence indicates the argument is the number format specification. 

The vv-list and number-format specification are \emph{trailing} optional
arguments. There is a possibility that the \verb`\eval` command could
be followed by a square-bracketed mathematical expression which \texttt{numerica}
might therefore confuse with one of its trailing arguments. Experience
using \texttt{numerica} suggests that this will be a (very) rare occurrence
and is easily prevented by inserting an empty brace pair (\verb`{}`)
before the offending square-bracketed expression. Allowing spaces
between the arguments enables complicated expressions and large vv-lists
to be formatted, in the interests of clarity, with new lines and white
space – without requiring the insertion of comment characters (\verb`%`).

Recommended practice is to minimise the number of optional arguments
used in \LaTeX{} commands by consolidating such arguments into a single
\emph{key=value} list. Although \texttt{numerica} uses such an argument,
the vv-list does not fit naturally into that scheme. And practice
suggests that separating out the elements of the number format specification
(rounding value, padding with zeros, scientific notation, boolean
output) and placing them in a trailing argument feels natural for
the kind of back-of-envelope calculations envisaged for \texttt{numerica}. 

\section{The variable=value list}

\label{subsec:evalVv-list}To evaluate algebraic, trigonometric and
other formulas that involve \emph{variables} we need to give those
variables values. This is done in the \emph{variable=value list} –
or \emph{vv-list} for short. This is the fourth argument of the \texttt{\textbackslash nmcEvaluate}
command and is a square-bracket delimited optional argument (optional
because an expression may depend only on constants and numbers).

\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
I was sorely tempted to use parentheses to delimit this argument,
since then both the placement and delimiters of the vv-list would
anticipate the way it is displayed in the evaluated result (see the
$mc^{2}$ example in §\ref{subsec:introSimple-examples} above). But
there is good reason not to. Parentheses will often occur in expressions
in the vv-list. With parentheses nested within parentheses it is all
too easy to get a pairing wrong, which would cause a \LaTeX{} error
and halt compilation. As it is, using the standard square bracket
delimiters, unbalanced parentheses cause a \texttt{numerica} error
(see §\ref{subsec:errorsMismatched-brackets}), which does not halt
compilation. (Of course unbalanced \emph{square} brackets now will
cause a \LaTeX{} error, but such brackets are used less often in mathematical
expressions and are rarely nested within other square-bracketed expressions.)\end{shaded}%
\end{minipage}

\subsection{Variable names}

\label{subsec:evalVariableNames}In mathematical practice, variable
names are generally single letters of the Roman or Greek alphabets,
sometimes also from other alphabets, in a variety of fonts, and often
with subscripts or primes or other decorations. In \texttt{numerica}
a variable name is \emph{what lies to the left of the equals sign
in an item} of the vv-list. Thus variables can be multi-token affairs:
$x',x'',x^{iv},x_{n},x'_{n},x''_{mn}$, $^{k}C_{n},var,\mathrm{var},Fred,\mathbf{Fred},\mathcal{FRED}\ldots$
Although variable names start and end with non-space tokens, a variable
name may contain spaces – for instance \verb`x x` should not cause
a \verb`numerica` error, but such names are not part of mathematical
practice. Usually, for the kind of back-of-envelope calculations envisaged
for \verb`numerica`, and for ease of typing, most variables will
be single letters from the Roman or Greek alphabets. 

Because equals signs and commas give structure to the vv-list, it
should also be clear that a variable name should not contain a \emph{naked}
equals sign or a \emph{naked} comma. They can be incorporated in a
variable name but only when decently wrapped in braces, like \verb`R_{=}`
displaying as $R_{=}$ or \verb`X_{,i}` displaying as $X_{,i}$. 

Note that $x$ and $\mathrm{x}$ will be treated by \verb`numerica`
as \emph{different} variables since, in the underlying \LaTeX , one
is \texttt{x} and the other \verb`\mathrm{x}`. Even names that look
identical in the pdf may well be distinct in \LaTeX . This is true
particularly of superscripts and subscripts: \verb`x_0` and \verb`x_{0}`
appear identical in the pdf but in the underlying \LaTeX{} they are
distinct, and will be treated as distinct variables by \verb`numerica`. 

Although multi-token variables are perfectly acceptable, \emph{internally}
\verb`numerica` expects variables to be single tokens. Hence a necessary
initial step for the package is to convert all multi-token variable
names in the vv-list and the formula to single tokens. \verb`numerica`
does this by turning the multi-token variable names into control sequences
with names in the sequence \verb`\_nmca`, \verb`\_nmcb`, \verb`\_nmcc`,
etc., then searches through the vv-list and the formula for every
occurrence of the multi-token names and replaces them with the relevant
control sequences. It does this in order of decreasing size of name,
working from the names that contain most tokens down to names containing
only two tokens. 

The conversion process uses computer resources. Even if there are
no multi-token variables present, \verb`numerica` still needs to
check that this is so – unless the user alerts the program to the
fact. This can be done by making a brief entry \texttt{xx=0 }in the
settings option (the second optional argument of \verb`\nmcEvaluate`);
see §\ref{subsec:settingsMultitokSwitch}. If the user never (or hardly
ever) uses multi-token variables, then a more permanent solution is
to create a file \verb`numerica.cfg`\texttt{ }with the line \texttt{multitoken-variables
= false}; see §\ref{sec:settingsDefaults} for this.

\subsection{The vv-list and its use}

A vv-list is a comma-separated list where each item is of the form
\emph{variable=value}. It might be something simple like
\begin{lyxcode}
{[}g=9.81,t=2{]}
\end{lyxcode}
or something more complicated like
\begin{lyxcode}
{[}V\_S=\textbackslash tfrac43\textbackslash pi~r\textasciicircum 3,V\_C=2\textbackslash pi~r\textasciicircum 2h,h=3/2,r=2{]}.
\end{lyxcode}
Spaces around the equals signs or the commas are stripped away during
processing so that 
\begin{lyxcode}
{[}g=9.81,t=2{]}~\textrm{and}~{[}~g~=~9.81~,~t~=~2{]}
\end{lyxcode}
are the \emph{same} variable=value list.

\subsubsection{Evaluation from right to left}

In these examples, with variables depending on other variables, there
is an implication: that the list is evaluated \emph{from the right}.
Recall how a function of a function is evaluated, say\emph{ }$y=f(g(h(x)))$.
To evaluate $y$, first $x$ is assigned a value then $h(x)$ is calculated,
then $g(h(x))$ then $f(g(h(x)))=y$. We work from right to left,
from the innermost to the outermost element. Or consider an example
like calculating the area of a triangle by means of the formula
\[
A=\sqrt{s(s-a)(s-b)(s-c)}.
\]
First we write the formula; then we state how $s$ depends on $a,b,c$,
namely $s=\frac{1}{2}(a+b+c)$, then we give values to $a,b,c$. In
\texttt{numerica} this is mirrored in the layout of the \verb`\eval`
command:
\begin{verbatim}
    \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
                 [s=\tfrac12(a+b+c),a=3,b=4,c=5]
\end{verbatim}
The formula in a sense is the leftmost extension of the vv-list. The
entire evaluation occurs from right to left.

This means that the rightmost variable in the vv-list can depend only
on (mathematical) constants and numbers – although it may be a complicated
expression of those elements. Other variables in the vv-list can depend
on variables \emph{to their right} but not to their left.

\subsubsection{Expressions in the variable=value list}

Suppose our expression is $\tfrac{4}{3}\pi r^{3}$, the volume $V_{S}$
of a sphere in terms of its radius $r$, and we want to calculate
the volume for different values of $r$ to get a sense of how rapidly
volume increases with radius.
\begin{centred}
\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $` $\Longrightarrow$ $ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $.
\end{centred}
Having set up this calculation it is now an easy matter to change
the value of $r$ in the vv-list:
\begin{centred}
\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1.5] $` $\Longrightarrow$
$ V_S= \eval{ \tfrac43\pi r^3 }[r=1.5] $.

\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=2] $` $  \Longrightarrow $ $V_S= \eval{ \tfrac43\pi r^3 }[r=2] $.
\end{centred}
To compute the volume $V_{C}=\pi r^{2}h$ of a cylinder, we have two
variables to assign values to:
\begin{centred}
\verb`$ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $` $\Longrightarrow$ $ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $.
\end{centred}
Although values in the vv-list are generally either numbers or simple
expressions (like \texttt{4/3}), that is not essential. A little more
complicated is
\begin{centred}
\verb`$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $.
\end{centred}
where calculation of the volume of the cylinder has been split into
two: first calculate the area $A_{C}$ of its circular base and then,
once that has been effected, calculate the volume.

A second example is provided by Brahmagupta's formula for the area
of a triangle in terms of its semi-perimeter. In a triangle ABC, the
sides are $a=3$, $b=4$ and $c=5$. (Of course we know this is a
right-angled triangle with area $\tfrac{1}{2}ab=6$.) The semi-perimeter
$s=\tfrac{1}{2}(a+b+c)$ and the area of ABC is \medskip{}
\begin{verbatim}
    \eval{$ \sqrt{s(s-a)(s-b)(s-c) $}
                 [s=\tfrac12(a+b+c),a=3,b=4,c=5] 
\end{verbatim}
$\Longrightarrow$ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
    [s=\tfrac12(a+b+c),a=3,b=4,c=5].

\subsubsection{Constants}

There are five constants built-in to \texttt{numerica}: \verb`\pi`,
the ratio of circumference to diameter of a circle, \verb`e`, the
base of natural logarithms, \verb`\gamma`, the limit of $\left(\sum_{1}^{N}1/n\right)-\ln N$
as $N\to\infty$, \verb`\phi`, the golden ratio, equal to $\tfrac{1}{2}(1+\surd5)$,
and the utilitarian constant \verb`\deg`, the number of radians in
a degree.
\begin{centred}
\verb`\eval{$ \pi $}` $\Longrightarrow$ \eval{$ \pi $},

\verb`\eval{$ e $}` $\Longrightarrow$ \eval{$ e $},

\verb`\eval{$ \gamma $}` $\Longrightarrow$ \eval{$ \gamma $},

\verb`\eval{$ \phi $}` $\Longrightarrow$ \eval{$ \phi $},

\verb`\eval{$ \deg $}` $\Longrightarrow$ \eval{$ \deg $},
\end{centred}
so that \verb`\eval{$ 180\deg $}` $\Longrightarrow$ \eval{$ 180\deg $}
(as it should).

Let's combine two of these in a formula:
\begin{centred}
\verb`\eval{$ e^\pi-\pi^e $}` $\Longrightarrow$ \eval{$ e^\pi-\pi^e $},
\end{centred}
which is close-ish to $\tfrac{1}{4}e$: \verb`\eval{$ \tfrac14e $}`
$\Longrightarrow$ \eval{$ \tfrac14e $}.

\subsubsection{Use of \texttt{\textbackslash pi},\texttt{ e},\texttt{ \textbackslash gamma},\texttt{
\textbackslash phi} as variables}

In some contexts it may feel natural to use any or all of \verb`\pi`,
\verb`e`, \verb`\gamma` and \verb`\phi` as variables by assigning
values to them in the vv-list. \texttt{numerica} does not object.
The values assigned in this way override the constants' values. For
example, if the triangle we labelled ABC previously was instead labelled
CDE then it has sides $c=3,d=4$ and (note!) $e=5$. It's area therefore
is\medskip{}
\begin{verbatim}
    \eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
              [s=\tfrac12(c+d+e),c=3,d=4,e=5]       
\end{verbatim}
$\Longrightarrow$ 

\eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
    [s=\tfrac12(c+d+e),c=3,d=4,e=5].\medskip{}

\noindent Since this is the correct area we see that \texttt{e} has
been treated as a variable with the assigned value $5$, not as the
constant. But if \texttt{e} (or \verb`\pi` or \verb`\gamma` or \verb`\phi`)
is not assigned a value in the vv-list then it has, by default, the
value of the constant. 

In the case of \texttt{e}, if you wish to use it as a variable, the
constant is always available as \verb`\exp(1)`. There is no similar
alternative available for \verb`\pi`, \verb`\gamma` or \verb`\phi`
although you can always do something like \verb`[\pi=<new value>,\pi_0=\pi]`
in the vv-list, so that \verb`\pi_0` now has the constant's value.

\subsection{Display of the vv-list}

By default, the vv-list is displayed with (in fact following) the
numerical result. That and the format of the display can both be changed.

\subsubsection{Star option: suppressing display of the vv-list}

\label{subsec:evalVvSuppresList}If display of the vv-list is not
wanted at all, only the numerical result, it suffices to attach an
asterisk (star) to the \texttt{\textbackslash eval} command:
\begin{centred}
\verb`$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $,
\end{centred}
or simply the naked result:
\begin{centred}
\verb`\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1]` $\Longrightarrow$
\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1].
\end{centred}
In the latter case, note that a negative result will display with
a hyphen for the minus sign unless you, the user, explicitly write
math delimiters around the \verb`\eval*` command:
\begin{centred}
\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]
\end{centred}
The \verb`$` signs that \verb`\eval*` wraps around are ignored.
The star option delivers a number, pure and simple, with no accompaniments.

\subsubsection{Suppressing display of items}

\label{subsec:evalVvSuppressVars}You may wish to retain some variables
in the vv-list display, but not all. For those variables you wish
omitted from the display, wrap each variable (but not the equals sign
or value) in braces. When calculating the volume of a cylinder in
the previous examples, the base area $A_{C}$ has a different status
from the `fundamental' variables $r$ and $h$. It is an intermediate
value, one that we pass through on the way to the final result. To
suppress it from display enclose the variable in braces:
\begin{centred}
\verb`$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $.
\end{centred}
As you can see, $A_{C}$ no longer appears in the displayed vv-list.
Of course the name and its value are still recorded `behind the scenes'
and can still be used in calculations.

\subsubsection{Changing the display format}

\label{subsec:evalChanging-display-format}In two examples above,
we have calculated the area of a triangle using Brahmagupta's formula.
Display of the result is crowded. Two remedies have just been suggested,
but a third one and preferable in this case would be to force display
of the vv-list and result to a new line. This can be done through
the settings option to the \verb`\eval` command, discussed in §\ref{subsec:settings New-line-display}.
However, if \verb`\eval` is wrapped around an \emph{appropriate}
environment (like \verb`multline`, but not \verb`equation`) it can
also be done simply by including \texttt{\textbackslash\textbackslash}
at the end of the formula. 

In the following example I use Brahmagupta's formula for calculating
the area of a cyclic quadrilateral (of which his formula for a triangle
is a special case). The cyclic quadrilateral in the example is formed
by a 45-45-90 triangle of hypotenuse 2 joined along the hypotenuse
to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
Adding the areas of the two triangles, the area of the quadrilateral
is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
formula:
\begin{verbatim}
    \eval{
      \begin{multline*}
        \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
      \end{multline*} 
         }[s=\tfrac12(a+b+c+d),
           a=\surd2,b=\surd2,c=\surd3,d=1] 
\end{verbatim}
$\Longrightarrow$     \eval{
      \begin{multline*}
        \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
      \end{multline*} 
         }[s=\tfrac12(a+b+c+d),
           a=\surd2,b=\surd2,c=\surd3,d=1]

\section{Formatting the numerical result}

\label{subsec:evalRoundingEtc} A result of a calculation is displayed,
by default, to $6$ decimal places. All our results so far have been
rounded to this figure, although not all digits are displayed, for
instance if the sixth one is $0$, or the result is an integer. Like
other elements of the display, both rounding value and the (dis)appearance
of trailing zeros can be customized, in this case by means of an optional
argument following the vv-list (or the formula if there is no vv-list).
This optional argument may contain up to four juxtaposed items from
seven possibilities:
\begin{itemize}
\item a question mark ?, which gives boolean output, or 
\item an integer, the \emph{rounding value}, positive, negative or zero,
specifying how many decimal places to display the result to, or
\item an asterisk {*}, which pads the result with zeros should it not have
as many decimal places as the rounding value specifies, or
\item the character \texttt{x} (lower case!) which presents the result in
`proper' scientific notation (a form like $1.234\times10^{5}$ for
123450), or
\item the character \texttt{t} (lower case!) which presents the result in
a bastardized form of scientific notation useful in tables (a form
like $(5)1.234$ for 123450), or
\item a character other than \texttt{?}, \texttt{{*}}, \texttt{x}, \texttt{t}
or a digit, usually (but not necessarily) one of the letters\texttt{
e d} \texttt{E D}, which presents the result in scientific notation
using that character as the exponent mark (a form like $1.234\text{e}5$
for $123450$), or
\item a prime 
\begin{itemize}
\item attached to the character specifying scientific notation, which extends
that notation to numbers in the interval \texttt{{[}1,10)}, or
\item attached to a question mark, which changes the format of boolean output.
\end{itemize}
\end{itemize}
If you use \texttt{?} in the same specification as some other text
character, the \texttt{?} prevails; if you use \texttt{x} in the same
specification as some other text character except for \texttt{?},
the \texttt{x} prevails; if you use \texttt{t} in the same specification
as some other text character except for \texttt{?} or \texttt{x},
the \texttt{t} prevails.

\subsection{Rounding value }

\label{subsec:evalRounding-value}The rounding value specifies the
number of decimal places displayed:
\begin{centred}
\verb`$ \eval{ 1/3 }[4] $` $\Longrightarrow$ $ \eval{ 1/3 }[4] $
\end{centred}
The default rounding value is $6$:
\begin{centred}
\verb`$ \eval{ 35/3 } $` $\Longrightarrow$ $ \eval{ 35/3 } $
\end{centred}
Following the default behaviour in \texttt{l3fp}, `ties' are rounded
to the nearest \emph{even} digit. Thus a number ending $55$ has a
`choice' of rounding to $5$ or $6$ and rounds up to the even digit
$6$, and a number ending $65$ with a `choice' of rounding to $6$
or $7$ rounds down to the even digit $6$:
\begin{centred}
\verb`$ \eval{ 0.1234555 } $`  $\Longrightarrow\eval{0.1234555}$ 

\verb`$ \eval{ 0.1234565 } $`  $\Longrightarrow\eval{0.1234565}$ 
\end{centred}
The calculational engine which \texttt{numerica} uses, \texttt{l3fp},
works to 16 significant figures and never displays more than that
number (and often less). 
\begin{itemize}
\item In the first of the following although I have specified a rounding
value of $19$ only $16$ decimal places are displayed, with the final
digit rounded up to $7$; 
\item in the second I have added $10$ zeros after the decimal point, meaning
that all $19$ decimal places specified by the rounding value can
be displayed since the 10 initial zeros do not contribute to the significant
figures; 
\item in the third I have changed the figure \emph{before} the decimal point
to $1$ so that the $10$ added zeros are now included among the significant
figures; 
\item and in the fourth, I have added $9$ digits before the decimal point:
\end{itemize}
\begin{centred}
\verb`$ \eval{ 0.1234567890123456789 }[19] $` $\Longrightarrow$
$\eval{ 0.1234567890123456789 }[19]$

\verb`$ \eval{ 0.00000000001234567890123456789 }[19] $` $\Longrightarrow$
$\eval{ 0.00000000001234567890123456789 }[19]$

\verb`$ \eval{ 1.00000000001234567890123456789 }[19] $` $\Longrightarrow$
$\eval{ 1.00000000001234567890123456789 }[19]$

\verb`$ \eval{ 987654321.1234567890123456789 }[19] $` $\Longrightarrow$
$\eval{ 987654321.1234567890123456789 }[19]$
\end{centred}
In all cases, no more than $16$ \emph{significant} figures are displayed,
although the number of decimal places displayed may exceed $16$ as
in the second example.

It is possible to use \emph{negative} rounding values. Such a value
zeroes the specified number of digits \emph{before} the decimal point.
\begin{centred}
\verb`$ \eval{ 987654321.123456789 }[-4] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[-4]$
\end{centred}
A rounding value of $0$ rounds to the nearest integer:
\begin{centred}
\verb`$ \eval{ 987654321.123456789 }[0] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[0]$
\end{centred}
If you wish to change the \emph{default} rounding value from $6$
to some other value, this can be done by creating or editing a file
\texttt{numerica.cfg} in a text editor; see §\ref{sec:settingsDefaults}.

\subsection{Padding with zeros }

\label{subsec:evalPadding-with-zeros}A result may contain fewer decimal
places than the rounding value specifies, the trailing zeros being
suppressed by default (this is how \texttt{l3fp} does it). Sometimes,
perhaps for reasons of presentation like aligning columns of figures,
it may be desirable to pad results with zeros. This is achieved by
inserting an asterisk, {*}, into the final optional argument of the
\verb`\eval` command:
\begin{centred}
\verb`$ \eval{ 1/4 }[4] $` $\Longrightarrow$ $ \eval{ 1/4 }[4] $,

\verb`$ \eval{ 1/4 }[4*] $` $\Longrightarrow$ $ \eval{ 1/4 }[4*] $.
\end{centred}

\subsection{Scientific notation }

\label{subsec:evalScientificNotation} The \texttt{l3fp} package can
output numbers in scientific notation. For example, $1234$ is rendered
as $\eval{1234}[e]$, denoting $1.234\times10^{3}$ , and $0.008$
as $\eval{0.008}[e]$, denoting $8\times10^{-3}$. The `e' here,
the \emph{exponent mark}, separates the \emph{significand} ($1.234$)
from the \emph{exponent} ($3$). To switch on output in scientific
notation in \texttt{numerica} enter \texttt{e} in the trailing optional
argument: 
\begin{centred}
\verb`$ \eval{ 123456789 }[e] $` $\Longrightarrow$  $ \eval{ 123456789 }[e] $. 
\end{centred}
The default rounding value $6$ is in play here. In \texttt{numerica},
when scientific notation is selected rounding takes a different meaning:
it is the \emph{significand }which is rounded (not the number as a
whole). One digit precedes the decimal point, at most $15$ follow
it. 

Negative rounding values are pointless for scientific notation. A
zero might on occasion be relevant:
\begin{centred}
\verb`$ \eval{ 987654321 }[0e] $` $\Longrightarrow$ $ \eval{ 987654321 }[0e] $.
\end{centred}
Sometimes letters other than `e' are used to indicate scientific
notation, like `E' or `d' or `D'. With a few exceptions, \texttt{numerica}
allows any letter or text character to be used as the exponent marker:
\begin{centred}
\verb`\eval{$ 1/23456789 $}[4d]`\texttt{ $\Longrightarrow$} \eval{$ 1/23456789 $}[4d].
\end{centred}
But when \texttt{x} is inserted in the trailing optional argument,
the output is in the form $d_{0}.d_{1}\ldots d_{m}\times10^{n}$ (except
when $n=0$), where each $d_{i}$ denotes a digit.
\begin{centred}
\verb`\eval{$ 1/23456789 $}[4x]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4x] .
\end{centred}
The requirements of tables leads to another form of scientific notation.
Placing \texttt{t} in the trailing argument turns on this table-ready
form of notation:
\begin{centred}
\verb`\eval{$ 1/23456789 $}[4t]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4t].
\end{centred}
This is discussed more fully in the associated document \texttt{numerica-tables.pdf}.

In the next example three options are used in the trailing argument.
The order in which the items are entered does not matter: 
\begin{centred}
\verb`\eval{$ 1/125 $}[*e4]`  $\Longrightarrow$ \eval{$ 1/125 $}[*e4].
\end{centred}
Finally, to illustrate that `any' text character\footnote{Be sensible! An equals sign for instance might confuse \texttt{numerica}
into thinking the number-format option is the vv-list, and will certainly
confuse the reader.} save for \texttt{x} or \texttt{t} can be used to distinguish the
exponent, I use an @ character:
\begin{centred}
\verb`\eval{$ 1/123 $}[@4]` $\Longrightarrow$ \eval{$ 1/123 $}[@4].
\end{centred}

\subsubsection{Numbers in \texttt{{[}1,10)}}

Usually when scientific notation is being used, numbers with magnitude
in the interval $[1,10)$ are rendered in their normal decimal form,
$3.14159$ and the like. Occasionally it may be desired to present
numbers in this range in scientific notation (this can be the case
in tables where the alignment of a column of figures might be affected).
\texttt{numerica} offers a means of extending scientific notation
to numbers in this range by adding a prime to the letter chosen as
the exponent mark in the trailing optional argument.
\begin{centred}
\verb`\eval{$ \pi $}[4t']` $\Longrightarrow$ \eval{$ \pi $}[4t'] 
\end{centred}

\subsubsection{\textbackslash eval{*} and scientific notation}

Scientific notation can be used for the numerical result output by
\verb`\eval*`:
\begin{centred}
\verb`\eval*{$ \pi $}[e']` $\Longrightarrow$ \eval*{$ \pi $}[e'] 
\end{centred}
There is one catch: if you substitute \texttt{x} for \texttt{e} here,
\LaTeX{} will complain about a missing \verb`$`. An \texttt{x} in
the number-format option produces a \verb`\times` in the output which
requires a math environment. It is up to you, as the user, to provide
the necessary delimiters outside the \verb`\eval*` command. (This
applies even when \verb`\eval*` wraps around math delimiters.)

(Because of the way \texttt{numerica} parses the number-format option,
entering a prime with neither exponent character nor question mark
specified will result in scientific output using \texttt{e} as the
exponent mark. The last example could have been written \verb`\eval*{$ \pi $}[']`.)

\subsection{Boolean output}

\label{subsec:evalBoolean-output}\texttt{l3fp} can evaluate comparisons,
outputting $0$ if the comparison is false, $1$ if it is true. By
entering a question mark, \texttt{?}, in the trailing optional argument,
you can force \texttt{numerica} to do the same depending as the result
of a calculation is zero or not. The expression being evaluated does
not need to be a comparison, \verb`$ \eval{\pi}[?] $` $\Longrightarrow$ $ \eval{\pi}[?]$,
but comparisons are what this is designed for.

Possible comparison relations are \verb`=`, \verb`<`, \verb`>`,
\verb`\ne`, \verb`\neq`, \verb`\ge`, \verb`\geq`, \verb`\le`,
\verb`\leq`. Although programming languages use combinations like
\verb`<=` or \verb`>=`, \texttt{numerica} does \emph{not} accept
these (they are not part of standard \emph{mathematical} usage) and
will generate an error. An example where the relation is equality
exhibits a numerological curiosity:\footnote{The \texttt{{[}p=.{]}} of this and the next example ensures a full
stop appears in the correct place; see §\ref{subsec:settingsPunctuation}.}
\begin{centred}
\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]` $\Longrightarrow$
\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]
\end{centred}
Notice the $5$ alongside the question mark in the trailing argument.
That is critical. Change the $5$ to a $6$ (or omit it since the
default rounding value is $6$) and the outcome is different: 
\begin{centred}
\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]` $\Longrightarrow$
\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]
\end{centred}
Now the relation is false. Evaluating the fraction to more than $6$
places, say to $9$, we can see what is going on:
\begin{centred}
\verb`\eval{$ 1/0.0123456789 $}[9]` $\Longrightarrow$ \eval{$ 1/0.0123456789 $}[9]. 
\end{centred}

\subsubsection{Outputting \texttt{T} or \texttt{F}}

To my eye, outputting $0$ or $1$ in response to a `question' like
$1/0.0123456789=81$ is confusing. It is easy to change the boolean
output from $0,1$ to a more appropriate $F,T$, or \texttt{$\texttt{F,\texttt{T}}$}
by adding a prime or two primes respectively to the question mark
in the number-format option.
\begin{centred}
\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']` $\Longrightarrow$
\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']
\end{centred}
The default boolean output format is chosen to be $0,1$ in case an
\verb`\eval*` command is used within another \verb`\eval` command
(`nesting'– see Chapter~\ref{chap:Nesting}~). The inner command
needs to output a \emph{numerical} answer.

\subsubsection{Rounding error tolerance}

\label{subsec:evalToleranceRounding}If at least one of the terms
in a comparison is the result of a calculation, then it's value is
likely to contain rounding errors. What level of rounding error can
we tolerate before such errors interfere with the comparison being
made? \texttt{l3fp} tolerates none. It decides the truth or falsity
of a comparison to all $16$ significant figures: 1.000 0000 0000
0000 and 1.000 0000 0000 0001 are \emph{not }equal in \texttt{l3fp}.
But for most purposes this will be far too severe a criterion.

Suppose our comparison relation is $\varrho$, denoting one of =,
<, >, \verb`\le`, etc. If $X\rel Y$ then $X-Y\rel Y-Y$, i.e. $X-Y\rel0$.
This is what \texttt{numerica} does. It takes the right-hand side
of the relation from the left-hand side and then compares the \emph{rounded}
difference under $\varrho$ to $0$. The rounding value used is the
number specified with the question mark in the trailing argument of
the \verb`\eval` command or, if no number is present, the default
rounding value (`out of the box' this is $6$). Thus, in a recent
example, $1/0.0123456789-81$ when rounded to $5$ decimal places
is \@.$0.00000$, indistinguishable from zero at this rounding value;
hence the equality $1/0.0123456789=81$ is true. But when rounded
to $6$ places it is $0.000001$ which \emph{is} distinguishable from
zero and so the equality is false. Truth or falsity depends on the
rounding value. 

When dealing with numbers generated purely mathematically, rounding
values of $5$ or $6$ are likely to be too small. More useful would
be rounding values closer to \texttt{l3fp}'s $16$ – perhaps $14$?
– depending on how severe the calculations are that generate the numbers.
However if the numbers we are dealing with come from outside mathematics,
from practical experiments perhaps, then even a rounding value of
$5$ or $6$ may be too large. 

\subsubsection{Rationale}

Mathematically, the claim that $X=Y$ at a rounding value $n$ is
the claim that 
\[
\abs{X-Y}\le5\times10^{-(n+1)}.
\]
since this rounds \emph{down} to zero at $n$ places of decimals.
This gives a more accurate test of equality than doing things in the
opposite order – rounding each number first and then taking the difference.
One might, for instance, have numbers like $X=0.12345$, $Y=0.12335$.
Rounding to $n=4$ places, both round to $0.1234$ and yet the difference
between them is $0.0001$ – they are distinguishable numbers to $4$
places of decimals. This is why \texttt{numerica} forms the difference
\emph{before }doing the rounding.

\subsubsection{And, Or, Not}

For logical And \LaTeX{} provides the symbols \verb`\wedge` and \verb`\land`,
both displaying as $\land$, but \texttt{numerica} adds thin spaces
( \verb`\,` ) around the symbol for \verb`\land` (copying the package
\texttt{gn-logic14.sty}). For logical Or \LaTeX{} provides the symbols
\verb`\vee` and \verb`\lor`, both displaying as $\lor$, but again
\texttt{numerica} adds thin spaces around the symbol for \verb`\lor`. 
\begin{centred}
\verb`\eval{$ 1<2 \wedge 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \wedge 2<3 $}[?''],

\verb`\eval{$ 1<2 \land 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \land 2<3 $}[?''].
\end{centred}
To my eye the second of these with its smidgen more space around the
wedge symbol displays the meaning of the overall expression better
than the first. Both And and Or have equal precedence; in cases of
ambiguity the user needs to parenthesize as necessary to clarify what
is intended.

\LaTeX{} provides two commands for logical Not, \verb`\neg` and \verb`\lnot`,
both displaying as $\lnot$ . Not binds tightly to its argument:
\begin{centred}
\verb`\eval{$ \lnot A \land B $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot A \land B $}[A=0,B=0].
\end{centred}
Here \verb`\lnot` acts only on the $A$; if it had acted on $A\land B$
as a whole the result would have been $1$.

For a little flourish, I evaluate a more complicated logical statement:\footnote{Quoting from an article in \emph{Quanta Magazine} (August 2020) by
Kevin Hartnett: `Let’s say you and two friends are planning a party.
The three of you are trying to put together the guest list, but you
have somewhat competing interests. Maybe you want to either invite
Avery or exclude Kemba. One of your co-planners wants to invite Kemba
or Brad or both of them. Your other co-planner, with an ax to grind,
wants to leave off Avery or Brad or both of them. Given these constraints,
you could ask: Is there a guest list that satisfies all three party
planners?' I have written $C$ for Kemba, $A$ and $B$ for Avery
and Brad.}
\begin{verbatim}
    \eval{$(A\lor\lnot C)\land(C\lor B)\land
                (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']
\end{verbatim}
$\Longrightarrow$ \eval{$(A\lor\lnot C)\land(C\lor B)\land
                (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']

\subsubsection{Chains of comparisons}

\texttt{numerica} can handle chains of comparisons like $1<2<1+2<5-1$.
`Behind the scenes' it inserts logical And-s into the chain, $1<2\land2<1+2\land1+2<5-1$,
and evaluates the modified expression:
\begin{centred}
\verb`\eval{$ 1<2<1+2<5-1 $}[?'']` $\Longrightarrow$ \eval{$ 1<2<1+2<5-1 $}[?''].
\end{centred}

\subsubsection{\texttt{amssymb} comparison symbols}

\label{subsec:evalAmssymb-comparisons}\texttt{numerica} accepts some
alternative symbols for the basic comparison relations from the \texttt{amssymb}
package provided that package is loaded, i.e. the preamble of your
document includes the statement
\begin{lyxcode}
\textbackslash usepackage\{amssymb\}
\end{lyxcode}
The variants from this package are: \verb`\leqq` ( $\leqq$ ), \verb`\leqslant`
( $\leqslant$ ), \verb`\geqq` (~$\geqq$~), and \verb`\geqslant`
( $\geqslant$ ).\footnote{No, that is not \texttt{eggplant}.} There
are also negations: \verb`\nless` ( $\nless$ ), \verb`\nleq` (~$\nleq$~),
\verb`\nleqq` ( $\nleqq$ ), \verb`\nleqslant` ( $\nleqslant$ ),
\verb`\ngtr` ( $\ngtr$ ), \verb`\ngeq` ( $\ngeq$ ), \verb`\ngeqq`
( $\ngeqq$ ), \verb`\ngeqslant` ( $\ngeqslant$ ).

\section{Calculational details}

\subsection{Arithmetic}

\label{subsec:evalArithmetic}Addition, subtraction, multiplication,
division, square roots, \emph{$n$}-th roots, and exponentiating (raising
to a power) are all available. 

Multiplication can be rendered explicitly with an asterisk, 
\begin{centred}
\verb`\eval{$ 9*9 $}` $\Longrightarrow$ \eval{$ 9*9 $},
\end{centred}
but that's ugly. More elegant is to use \verb`\times`:
\begin{centred}
\verb`\eval{$ 9\times9 $}` $\Longrightarrow$ \eval{$ 9\times9 $}.
\end{centred}
\verb`\cdot` is also available and in many cases juxtaposition alone
suffices: 
\begin{centred}
\verb`\eval{$ \surd2\surd2 $}` $\Longrightarrow$ \eval{$ \surd2\surd2 $},

\verb`\eval{$ ab $}[a=123,b=1/123]` $\Longrightarrow$ \eval{$ ab $}[a=123,b=1/123].
\end{centred}
Division can be rendered in multiple ways too:
\begin{centred}
\verb`\eval{$ 42/6 $}` $\Longrightarrow$ \eval{$ 42/6 $},

\verb`\eval{$ 42\div6 $}` $\Longrightarrow$ \eval{$ 42\div6 $},
\end{centred}
or by using \verb`\frac` or \verb`\tfrac` or \verb`\dfrac` as in
\begin{centred}
\verb`\eval{$ \frac{42}6 $}` $\Longrightarrow$ \eval{$ \frac{42}6 $}.
\end{centred}
But note that since juxtaposition means multiplication, it is also
true that $42\tfrac{1}{6}$\texttt{ }evaluates to $7$ inside an \verb`\eval`
command rather than denoting `forty two and a sixth'. Hence if you
want to use `two and a half' and similar values in \texttt{numerica},
they need to be entered as improper fractions like $\tfrac{5}{2}$
or in decimal form, $2.5$ (as one does automatically in mathematical
expressions anyway because of the ambiguity in a form like $2\tfrac{1}{2}$).

\subsubsection{Square roots and $n$-th roots}

\label{subsec:evalSquareRootsEtc}Let us check that 3, 4, 5 and 5,
12, 13 really are Pythagorean triples (I use \verb`\sqrt` in the
first, \verb`\surd` in the second):
\begin{centred}
\verb`\eval{$ \sqrt{3^2+4^2} $}` $\Longrightarrow$ \eval{$\sqrt{3^{2}+4^{2}}$},

\verb`\eval{$ \surd(5^2+12^2) $}` $\Longrightarrow$ \eval{$ \surd(5^2+12^2) $}.
\end{centred}
The \verb`\sqrt` command has an optional argument which can be used
for extracting $n$-th roots of a number. This notation is generally
used when $n$ is a small positive integer like $3$ or $4$. This
practice is followed in \texttt{numerica}: $n$ must be a (not necessarily
small) \emph{positive integer}:
\begin{centred}
\verb`\eval{$ \sqrt[4]{81} $}` $\Longrightarrow$  \eval{$ \sqrt[4]{81} $},

\verb`\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}]` $\Longrightarrow$
\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}].
\end{centred}
If $n$ should not be a positive integer, an error message is generated;
see §\ref{sec:evalErrors}.

For display-style expressions, the \verb`\sqrt` command grows to
accommodate the extra vertical height; the surd doesn't. Here is an
example which anticipates a number of matters not discussed yet. It
shows \verb`\eval` wrapping around a square root containing various
formatting commands (negative spaces, \verb`\left` and \verb`\right`
nested within \verb`\bigg` commands), all digested without complaint
(see §\ref{subsec:evalFormatting-commands}; and see §\ref{subsec:settingsPunctuation}
for the \verb`[p=.]`): \medskip{}
\begin{verbatim}
    \eval[p=.]
      {\[ 
        \sqrt[3]
          {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
      \]}[A=729,B=81,C=9,D=3]
\end{verbatim}
$\Longrightarrow$\eval[p=.]
  {\[ 
    \sqrt[3]
      {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
  \]}[A=729,B=81,C=9,D=3]

As implemented in \texttt{numerica}, $n$-th roots found using \verb`\sqrt[n]`
are \verb`n=<integer>`\emph{ }roots. This raises an interesting question:
if the `$n$' of an $n$-th root is the result of a calculation,
what happens with rounding errors? The calculation may not produce
an \emph{exact} integer. (This problem also arises with factorials;
see §\ref{subsec:evalFactorialsBinom}.) The solution employed in
\texttt{numerica} is to make what is considered an integer depend
on a rounding value. Most calculations will produce rounding errors
in distant decimal places. For `int-ifying' calculations, \texttt{numerica}
uses a rounding value of $14$: a calculation produces an integer
if, when rounded to $14$ figures, the result is an integer. Since
\texttt{l3fp} works to $16$ significant figures, a rounding value
of $14$ allows ample `elbowroom' for rounding errors to be accommodated
when judging what is an integer and what is not. As a practical matter
problems should not arise. 

\subsubsection{\emph{n}-th roots of negative numbers}

Odd (in the sense of `not even') integral roots of \emph{negative}
numbers are available with \verb`\sqrt`,
\begin{centred}
\verb`\eval{$ \sqrt[3]{-125} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-125} $},

\verb`\eval{$ \sqrt[3]{-1.25} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-0.125} $}.
\end{centred}

\subsubsection{Inverse integer powers }

Of course to find an $n$-th root we can also raise to the inverse
power,
\begin{centred}
\verb`\eval{$ 81^{1/4} $}` $\Longrightarrow$ \eval{$ 81^{1/4} $}.
\end{centred}
However, raising a \emph{negative} number to an inverse power generates
an error even when, mathematically, it should not. This matter is
discussed below in §\ref{subsec:errorsInverse-powers}.

\subsection{Precedence, parentheses}

The usual precedence rules apply: multiplication and division bind
equally strongly and more strongly than addition and subtraction which
bind equally stongly. Exponentiating binds most strongly. Evaluation
occurs from the left.
\begin{centred}
\verb`\eval{$ 4+5\times6+3 $}` $\Longrightarrow$ \eval{$ 4+5\times6+3 $},

\verb`\eval{$ 6\times10^3/2\times10^2 $}` $\Longrightarrow$ \eval{$ 6\times10^3/2\times10^2 $},
\end{centred}
which may not be what was intended. Parentheses (or brackets or braces)
retrieve the situation:
\begin{centred}
\verb`\eval{$ (4+5)(6+3) $}` $\Longrightarrow$ \eval{$ (4+5)(6+3) $},

\verb`\eval{$ (6\times10^3)/(2\times10^2) $}` $\Longrightarrow$
\eval{$ (6\times10^3)/(2\times10^2) $}.
\end{centred}
Because exponentiating binds most strongly, negative values must be
parenthesized when raised to a power. If not,
\begin{centred}
\verb`\eval{$ -4^2 $}` $\Longrightarrow$ \eval{$ -4^2 $},
\end{centred}
which is clearly not $(-4)^{2}$. But
\begin{centred}
\verb`\eval{$ (-4)^2 $}` $\Longrightarrow$ \eval{$ (-4)^2 $}.
\end{centred}

\subsubsection{Command-form brackets}

\label{subsec:evalCommandBrackets}Note that brackets of all three
kinds are available also in command form: \verb`\lparen \rparen`
(from \verb`mathtools`) for \verb`( )`, \verb`\lbrack \rbrack`
for \verb`[ ]`, and \verb`\lbrace \rbrace` for \verb`\{ \}`.

\subsection{Modifiers\texttt{ (\textbackslash left \textbackslash right}, etc.)}

The  \verb`\left` and \texttt{\textbackslash right} modifiers and
also the series of \verb`\big...` modifiers\texttt{ }(\texttt{\textbackslash}\verb`bigl \bigr`,
\verb`\Bigl \Bigr`, \verb`\biggl \biggr`, \verb`\Biggl \Biggr`)
are available for use with all brackets (parentheses, square brackets,
braces):
\begin{verbatim}
    \eval[p=.]{\[ \exp\left(
        \dfrac{\ln2}{4}+\dfrac{\ln8}{4} 
      \right) \]}
\end{verbatim}
$\Longrightarrow$ \eval[p=.]{\[ \exp\left( \dfrac{\ln2}{4}+\dfrac{\ln8}{4} \right) \]}

\texttt{numerica} also accepts their use with \texttt{.} (dot) and
with \texttt{/} (as noted earlier, the \verb`[p]` and \verb`[p=.]`
are explained at §\ref{subsec:settingsPunctuation}):
\begin{centred}
\verb`\eval[p]{\[ \left.\dfrac{3+4}{2+1}\right/\!\dfrac{1+2}{4+5} \]}`
$\Longrightarrow$ \eval[p=.]{\[ \left. \dfrac{3+4}{2+1} \right/\!\dfrac{1+2}{4+5} \]}
\end{centred}
They can be nested.

\subsection{Other formatting commands}

\label{subsec:evalFormatting-commands}There are many formatting commands
which change the layout of a formula on the page but do not alter
its content. These include various spacing commands like \verb`\!`,
\verb`\quad`, etc., phantoms (\verb`\phantom` etc.), \verb`\mathstrut`
from \TeX{} and its \verb`mathtools` cousin \verb`\xmathstrut`.

Consider the same package's \verb`\splitfrac` and \verb`\splitdfrac`.
The \verb`mathtools` documentation gives an example to illustrate
the use of these last two. I've mangled it to produce a ridiculous
illustration of their use, and of the modifiers \verb`\left \right`,
and of the command-form alternatives to parentheses \verb`\lparen \rparen`;
also the use of \verb`\dfrac`. A little mental arithmetic will convince
that we are evaluating the square root of $(9\times7)^{2}$ which
indeed is what we get:\footnote{For the \texttt{{[}p=.,vvd={]}} see §\ref{subsec:settingsPunctuation}
and §\ref{subsec:settingsVvDisplayChangeLocal}. The first puts the
concluding full stop in the right place; the second suppresses the
vv-list.} \medskip{}
\begin{verbatim}
	\eval[p=.,vvd=]{\[
    \sqrt{\left\lparen
      \frac{ \splitfrac{xy + xy + xy + xy + xy}
              {+ xy + xy + xy + xy}
           }
           { \dfrac z7}
          \right\rparen \left\lparen
            \frac{ \splitdfrac{xy + xy + xy + xy + xy}
                     {+ xy + xy + xy + xy}
                 }
                 {\dfrac z7}\right\rparen}
      \]}[x=2,y=5,z=10]
\end{verbatim}
$\Longrightarrow$\eval[p=.,vvd=]{\[
    \sqrt{\left\lparen
      \frac{ \splitfrac{xy + xy + xy + xy + xy}
              {+ xy + xy + xy + xy}
           }
           { \dfrac z7}
          \right\rparen \left\lparen
            \frac{ \splitdfrac{xy + xy + xy + xy + xy}
                     {+ xy + xy + xy + xy}
                 }
                 {\dfrac z7}\right\rparen}
      \]}[x=2,y=5,z=10]\texttt{numerica} essentially ignores formatting commands (the ones
it knows of). They do not alter the mathematical content of a formula,
only how it looks. But there will undoubtedly be formatting commands
it does not recognize which will probably trigger an `Unknown token'
message. Please contact the author in that case.\footnote{ajparsloe@gmail.com}

\subsection{Trigonometric \& hyperbolic functions}

\LaTeX{} provides all six trignometric functions, \verb`\sin`, \verb`\cos`,
\verb`\tan`, \verb`\csc`, \verb`\sec`, \verb`\cot` and the three
principal inverses \verb`\arcsin`, \verb`\arccos`, \verb`\arctan`.
It also provides four of the six hyperbolic functions: \verb`\sinh`,
\verb`\cosh`, \verb`\tanh`, \verb`\coth`, and \emph{no} inverses.
\texttt{numerica} provides the missing hyperbolic functions, \verb`\csch`
and \verb`\sech`, and all missing inverses, the three trigonometric
and all six hyperbolic: \verb`\arccsc`, \verb`\arcsec`, \verb`\arccot`,
and \verb`\asinh`, \verb`\acosh`, \verb`\atanh`, \verb`\acsch`,
\verb`\asech`, \verb`\acoth`. (\emph{HMF} writes $\text{arcsinh}$,
$\text{arccosh}$, etc. and ISO recommends $\text{arsinh}$, $\text{arcosh}$,
etc. The first seems ill-advised, the second not widely adopted. At
present neither is catered for in \texttt{numerica}.)\emph{ }
\begin{centred}
\verb`\eval{$ \arctan1/1\deg $}` $\Longrightarrow$ \eval{$ \arctan 1/1\deg $} ,

\verb`\eval{$ \atanh\tanh3 $}` $\Longrightarrow$ \eval{$ \atanh\tanh3 $} .
\end{centred}
Inverses can also be constructed using the `$-1$' superscript notation.
Thus
\begin{centred}
\verb`\eval{$ \sin^{-1}(1/\surd2)/1\deg $}` $\Longrightarrow$ \eval{$ \sin^{-1}(1/\surd2)/1\deg $} ,

\verb`\eval{$ \tanh\tanh^{-1}0.5 $}` $\Longrightarrow$ \eval{$ \tanh\tanh^{-1}0.5 $} .
\end{centred}
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\subsubsection*{Hyperbolic functions}

Please note that \texttt{l3fp} does not (as yet) provide \emph{any}
hyperbolic functions natively. The values \texttt{numerica} provides
for these functions are \emph{calculated} values using familiar formulas
involving exponentials (for the direct functions) and natural logarithms
and square roots for the inverses. Rounding errors mean the values
calculated may not have $16$-figure accuracy. The worst `offenders'
are likely to be the least used, \verb`\acsch` and \verb`\asech`.
For instance,
\[
\acsch x=\ln\left[\frac{1}{x}+\left(\frac{1}{x^{2}}+1\right)^{1/2}\right],
\]

\begin{centred}
\verb`\eval{$ \csch \acsch 7 $}[16]` $\Longrightarrow$ \eval{$ \csch \acsch 7 $}[16].
\end{centred}
\end{shaded}%
\end{minipage}

\subsection{Logarithms}

The natural logarithm \verb`\ln`, base $10$ logarithm \verb`\lg`,
and binary or base $2$ logarithm \verb`\lb` are all recognized,
as is \verb`\log`, preferably with a subscripted base:
\begin{centred}
\verb`\eval{$ \log_{12}1728 $}` $\Longrightarrow$ \eval{$ \log_{12}1728 $}
\end{centred}
If there is no base indicated, base $10$ is assumed. (The notations
\verb`\ln`, \verb`\lg`, and \verb`\lb` follow ISO 80000-2 recommendation,
which frowns upon the use of the unsubscripted \verb`\log` although
only \verb`\ln` appears widely used.) The base need not be explicitly
entered as a number. It could be entered as an expression or be specified
in the vv-list:
\begin{centred}
\verb`\eval*{$ \log_b c $}[b=2,c=1024]` $\Longrightarrow$ \eval*{$ \log_b c $}[b=2,c=1024],
\end{centred}
the log to base $2$ in this case. It is possible to use the unadorned
\verb`\log` with a base different from $10$; if you wish to do this
only for a particular calculation see §\ref{subsec:settingsLogBase},
or see §\ref{sec:settingsDefaults} if you want to make this default
behaviour.

\subsection{Other unary functions}

Other unary functions supported are the exponential function \verb`\exp`
and signature function \verb`\sgn` (equal to $-1$, $0$, or $1$
depending as its argument is $<0$, $=0$, or $>0$). 

\subsection{Squaring, cubing, \ldots unary functions}

\texttt{numerica} has no difficulty reading a familiar but `incorrectly
formed' expression like 
\[
\sin^{2}1.234+\cos^{2}1.234.
\]
You do not have to render it $(\sin1.234)^{2}+(\cos1.234)^{2}$ or
(heaven forbid) $(\sin(1.234))^{2}+(\cos(1.234))^{2}$. The everyday
usage is fine:
\begin{centred}
\verb`\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234]` $\Longrightarrow$
\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234] .
\end{centred}
Equally \texttt{numerica} has no difficulty reading the `correct'
but pedantic form
\begin{centred}
\verb`\eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234]`
$\Longrightarrow$ \eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234] .
\end{centred}
A hyperbolic identity is confirmed in this example:
\begin{centred}
\verb`\eval{$ \sinh 3x $}[x=1]` $\Longrightarrow$ \eval{$ \sinh 3x $}[x=1],\medskip{}

\verb`\eval{$ 3\sinh x+4\sinh^3x $}[x=1]` $\Longrightarrow$ \eval{$ 3\sinh x+4\sinh^3x $}[x=1].
\end{centred}
In fact all named unary functions in \texttt{numerica} can be squared,
cubed, etc., in this `incorrect' but familiar way, although the
practice outside the trigonometric and hyperbolic context seems (vanishingly?)
rare.

When the argument of the function is parenthesized and raised to a
power – like $\sin(\pi)^{2}$ – it is read by \texttt{numerica} as
the `sine of the square of pi', $\sin(\pi^{2})$, and \emph{not
}as the `square of the sine of pi', $(\sin\pi)^{2}$:
\begin{centred}
\verb`\eval{$ \sin(\pi)^2 $}` $\Longrightarrow$ \eval{$ \sin(\pi)^2  $} .
\end{centred}
Things are done like this in \texttt{numerica} above all to handle
the logarithm in a natural way. Surely $\ln x^{n}=n\ln x=\ln(x^{n})$
rather than $(\ln x)^{n}$? And if we wish to write (as we do) $\ln(1+1/n)^{n}=n\ln(1+1/n)=1-1/2n+1/3n^{2}-\ldots$
to study the limiting behaviour of $(1+1/n)^{n}$, then we cannot
avoid $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$.

\subsection{\emph{n}-ary functions}

The functions of more than one variable ($n$-ary functions) that
\texttt{numerica} supports are \verb`\max`, \verb`\min` and \verb`\gcd`,
greatest common divisor. The comma list of arguments to \verb`\max`,
\verb`\min` or \verb`\gcd` can be of arbitrary length. The arguments
themselves can be expressions or numbers. For \verb`\gcd`, non-integer
arguments are truncated to integers. Hence both $y$ and $3y$ are
independently truncated in the following example – to $81$ and $243$
respectively:
\begin{centred}
\verb`\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3]` $\Longrightarrow$
\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3] .
\end{centred}
(The truncation occurs in the argument of \verb`\gcd`, not in the
vv-list.) 

For $n$-ary functions, squaring, cubing, etc. follows a different
pattern from that for unary functions. For \verb`\max`, \verb`\min`,
\verb`\gcd` the argument of the function is a comma list. Squaring
the argument makes no sense. We understand the superscript as applying
to the function as a whole. (Consistency is not the point here; it
is what mathematicians do that \texttt{numerica} tries to accommodate.)
\begin{centred}
\verb`\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24]` $\Longrightarrow$
\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24] .
\end{centred}

\subsection{Delimiting arguments with brackets \& modifiers }

Arguments of unary and $n$-ary functions can be delimited not only
with parentheses, but also with square brackets and braces, both in
explicit character form and also in the command form of §\ref{subsec:evalCommandBrackets}.
The brackets, of whatever kind, can be qualified with \verb`\left \right`,
\verb`\bigl \bigr`, etc.\footnote{See §\ref{subsec:settingsPunctuation} for the \texttt{{[}p=.{]}}
(which ensures the concluding full stop appears in the correct place.} 
\begin{centred}
\verb`\eval[p=.]{\[ \sin\left\lbrack \dfrac\pi{1+2+3}\right\rbrack \]}`
$\Longrightarrow$\eval[p=.]{\[ \sin\left\lbrack\dfrac\pi{1+2+3}\right\rbrack \]}
\end{centred}

\subsection{Absolute value, floor \& ceiling functions}

It is tempting to use the \texttt{|} key on the keyboard for inserting
an absolute value sign. \texttt{numerica} accepts this usage, but
it is deprecated. The spacing is incorrect – compare $|-l|$ using
\texttt{|} against $\lvert-l\rvert$ using \verb`\lvert \rvert`.
Also, the identity of the left and right delimiters makes nested absolute
values difficult to parse. \texttt{numerica} does not attempt to do
so. Placing an absolute value constructed with \texttt{|} within another
absolute value constructed in the same way is likely to produce a
compilation error or a spurious result. \verb`\lvert \rvert` are
better in every way except ease of writing.\texttt{ }To aid such ease
\texttt{numerica }provides the \verb`\abs` function (using the \texttt{\textbackslash DeclarePairedDelimiter}
command of the \texttt{mathtools} package). This takes a mutually
exclusive star (asterisk) or square bracketed optional argument, and
a mandatory braced argument. The starred form expands to \verb`\left\lvert #1 \right\rvert`
where \verb`#1` is the mandatory argument:
\begin{centred}
\verb`\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]` $\Longrightarrow$
\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]
\end{centred}
The optional argument provides access to the \verb`\big...` modifiers:
\begin{verbatim}
    \eval[p=.]{\[
      \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}} 
    \]}[A=12,a=-10,C=7,c=-5] 
\end{verbatim}
$\Longrightarrow$     \eval[p=.]{\[ 
      \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}} 
    \]}[A=12,a=-10,C=7,c=-5] 

The form without either star or square bracket option dispenses with
the modifiers altogether:
\begin{centred}
\verb`\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].` $\Longrightarrow$
\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].
\end{centred}
As noted, the star and square bracketed option are mutually exclusive
arguments.

\texttt{numerica} also provides the functions \verb`\floor` and \verb`\ceil`,
defined in the same way, taking a mutually exclusive star or square
bracketed optional argument and for the starred forms expanding to
\verb`\left\lfloor #1 \right\rfloor` and \verb`\left\lceil #1 \right\rceil`
where \verb`#1` is the mandatory argument, and for the square bracket
option forms replacing the \verb`\left` and \verb`\right` with the
corresponding \verb`\big` commands. The form without star or square-bracket
option dispenses with any modifier at all.
\begin{centred}
\verb`\eval{$ \floor{-\pi} $}` $\Longrightarrow$ \eval{$ \floor{-\pi} $},

\verb`\eval{$ \ceil{\pi} $}` $\Longrightarrow$ \eval{$ \ceil{\pi} $}.
\end{centred}
The floor function, $\lfloor x\rfloor$, is the greatest integer $\le x$;
the ceiling function, $\lceil x\rceil$ is the smallest integer $\ge x$.
Like the absolute value, the floor and ceiling functions, can be nested: 
\begin{centred}
\verb`\eval{$ \floor{-\pi+\ceil{e}} $}` $\Longrightarrow$ \eval{$ \floor{-\pi+\ceil{e}} $}.
\end{centred}

\subsubsection{Squaring, cubing, \ldots{} absolute values, etc.}

These three functions can be raised to a power \emph{without} extra
parentheses:
\begin{centred}
\verb`\eval{$ \ceil{e}^2 $},` $\Longrightarrow$ \eval{$ \ceil{e}^2 $},

\verb`\eval{$ \abs{-4}^2 $}.` $\Longrightarrow$ \eval{$ \abs{-4}^2 $}.
\end{centred}

\subsection{Factorials, binomial coefficients}

\label{subsec:evalFactorialsBinom}Factorials use the familiar trailing
\texttt{!} notation: 
\begin{centred}
\verb`\eval{$ 7! $}` $\Longrightarrow$ \eval{$ 7! $},

\verb`\eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3]`
$\Longrightarrow$ \eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3].
\end{centred}
The examples illustrate how \texttt{numerica} interprets the argument
of the factorial symbol:\texttt{ }it `digests' 
\begin{enumerate}
\item a preceding (possibly multi-digit) integer, or
\item a preceding variable token, or
\item a bracketed expression, or
\item a bracket-like expression – an absolute value, floor or ceiling function,
\end{enumerate}
since they delimit arguments in a bracket-like way:
\begin{centred}
\verb`\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}` $\Longrightarrow$
\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}.
\end{centred}
The result of feeding the factorial an expression different in kind
from one of these four cases may give an error message or an unexpected
result. Use parentheses around such an expression; for example write
$(3^{2})!$, rather than $3^{2}!$.

Nesting of brackets for factorials is accepted:
\begin{centred}
\verb`\eval{$ ((5-2)!+1)! $}` $\Longrightarrow$ \eval{$ ((5-2)!+1)! $}.
\end{centred}
The factorials of negative integers or of non-integers are not defined
in \texttt{numerica}. Again there is the problem met in relation to
$n$-th roots of what happens if the argument of a factorial is the
result of a calculation and rounding errors mean it is not an exact
integer. This problem is unlikely to be of practical concern since
\texttt{numerica} rounds the result of such a calculation by default
to $14$ significant figures before offering it to the factorial.
Since \texttt{l3fp} works to $16$ significant figures, there is ample
`elbowroom' to accommodate rounding errors before the result of
a calculation ceases to round to an integer. 

\subsubsection{Double factorials}

The double factorial, written $n!!,$ is the product $n(n-2)(n-4)\ldots\times4\times2$
when $n$ is even and the product $n(n-2)(n-4)\ldots\times3\times1$
when $n$ is odd. 
\begin{centred}
\verb`\eval{$ 6!! $}` $\Longrightarrow$ \eval{$ 6!! $},

\verb`\eval{$ n!! $}[n=\sqrt{49}]` $\Longrightarrow$ \eval{$ n!! $}[n=\sqrt{49}],
\end{centred}
Since $n!=n!!(n-1)!!$ it follows that
\[
n!!=\frac{n!}{(n-1)!!}=\frac{(n+1)!}{(n+1)!!}.
\]
Putting $n=0$ in the outer equality shows that $0!!=1$. Now putting
$n=0$ in the left equality gives $(-1)!!=1$. Double factorials therefore
are defined for integers $\ge-1$.

\subsubsection{Binomial coefficients}

Binomial coefficients are entered in \LaTeX{} with the \verb`\binom`\textbf{
}command. It takes two arguments and has a text-style version \verb`\tbinom`
and a display-style version \verb`\dbinom`. As implemented in \texttt{numerica},
these are \emph{generalised} binomial coefficients:
\[
\binom{x}{k}=\frac{x(x-1)\dots(x-k+1)}{k(k-1)\dots1},\quad(x\in\mathbb{R},~k\in\mathbb{N}),
\]
where $x$ need not be a non-negative integer, and where $\binom{x}{0}=1$
by definition. Although the first (or upper) argument can be any real
number, the lower argument \emph{must} be a non-negative integer.
Thus, \verb`\eval{$ \tbinom53 $}` $\Longrightarrow$ \eval{$\tbinom53$},
\verb`\eval{$ \tbinom70 $}` $\Longrightarrow$ \eval{$\tbinom70$},
\verb`\eval{$ \tbinom{4.2}3 $}` $\Longrightarrow$ \eval{$\tbinom{4.2}3$},
but if the second (or lower) argument of \verb`\binom` is \emph{not}
a non-negative integer, \texttt{numerica} displays a message; see
§\ref{subsec:errorsIntegerArgs}. 

\subsection{Sums and products}

\texttt{numerica} recognizes sums (\verb`\sum` displaying as $\sum$)
and products (\verb`\prod` displaying as $\prod$), and expects both
symbols to have lower and upper summation/product limits specified.
The lower limit must be given in the form \emph{sum/prod variable
= initial value}; the upper limit requires only the final value to
be specified (although it can also be given in the form \emph{sum/prod
variable = final value}). The values may be expressions depending
on other variables and values but must evaluate to integers (or infinity
– see §\ref{sec:settingsInfiniteSumsProds}). Evaluating to an integer
means that they \emph{round} to an integer, using a rounding value
that is set by default to $14$; (recall that \texttt{l3fp} works
to $16$ significant figures). If a limit evaluates to a non-integer
at this `int-ifying' rounding value, an error message results. (To
change this `int-ifying' rounding value, see §\ref{subsec:defaultsIntifyingRounding}.)

As an example of expressions in the limits, this example uses the
floor and ceiling functions to convert combinations of constants to
integers (the \verb`[p]` is explained in §\ref{subsec:settingsPunctuation}),
\begin{centred}
\verb`\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}` $\Longrightarrow$\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}
\end{centred}
\noindent (which is $\sum_{n=1}^{9}n$).\emph{ }If the upper limit
is less than the lower limit the result is zero. Notice that there
is no vv-list. The summation variable does not need to be included
there unless there are other variables that depend on it. However,
in the case
\begin{centred}
\verb`\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]
\end{centred}
the upper limit $N$ is necessarily assigned a value in the vv-list. 

To the author it seems natural to enter the lower limit first, immediately
after the \verb`\sum` command (the sum is \emph{from }something \emph{to}
something), but no problem will accrue if the upper limit is placed
first (after all, the appearance of the formula in the pdf is the
same):
\begin{centred}
\verb`\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]
\end{centred}
Another example of a sum, using binomial coefficients this time, is 
\begin{centred}
\verb`\eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]`
$\Longrightarrow$ \eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]
\end{centred}
which is just \verb`\eval{$(x+y)^5$}[x=0.75,y=2.25]` $\Longrightarrow$
\eval{$ (x+y)^5 $}[x=0.75,y=2.25], or $3^{5}$.

Now let's calculate a product:
\begin{verbatim}
    \eval[p]{\[
      \prod_{k=1}^{100} 
        \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
          \]}[x=1][3]
\end{verbatim}
$\Longrightarrow$ \eval[p]{\[\prod_{k=1}^{100} \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)\]}[x=1][3]

\noindent to be compared with \verb`\eval{$ \sinh 1 $}[3]` $\Longrightarrow$
\eval{$ \sinh1 $}[3]. Obviously more terms than $100$ are required
in the product to achieve 3-figure accuracy. 

\subsubsection{Infinite sums and products}

There is a strong urge to use $\infty$ in the upper limit of this
product. Let's do so:
\begin{verbatim}
    \eval[p=.]{\[
      \prod_{k=1}^{\infty} 
        \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
          \]}[x=1][3] 
\end{verbatim}
$\Longrightarrow$ \eval[p=.]
  {\[
    \prod_{k=1}^{\infty} 
      \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
  \]}[x=1][3] 

\noindent Disappointingly, we still get the same result, deficient
by $1$ in the third decimal place. Obviously \texttt{numerica} has
not multiplied an infinite number of terms and, just as obviously,
the finite number of terms it \emph{has} multiplied are too few. How
\texttt{numerica} decides when to stop evaluating additional terms
in an infinite sum or product is discussed later, §\ref{sec:settingsInfiniteSumsProds}.

For this particular product the problem is that it converges slowly.
Any criterion for when to stop multiplying terms or, for an infinite
sum adding terms, seems bound to fail for some product or series.
Presumably any stopping criterion must measure smallness in some way.
But terms of the divergent harmonic series, $\sum(1/n)$ can always
be found smaller than any value we care to specify. It is not surprising
that a sufficiently slowly converging product or series falls foul
of a given criterion.

The default criterion however can be changed. Because this involves
values assigned in the settings option of the \verb`\eval` command,
I discuss infinite sums and products in the section discussing that
optional argument; see §\ref{sec:settingsInfiniteSumsProds}.

Other infinite sums converge more rapidly, and the default settings
work admirably. For example \verb`\eval{$ (1+0.1234)^{4.321} $}`
$\Longrightarrow$ \eval{$ (1+0.1234)^{4.321} $}. Using binomial
coefficients we can express this as an infinite sum:\medskip{}
\begin{verbatim}
    \eval[p=.]{\[
        \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
           \]}[\alpha=4.321,x=0.1234]
\end{verbatim}
$\Longrightarrow$ \eval[p=.]
  {\[
    \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
  \]}[\alpha=4.321,x=0.1234]

\section{Error messages }

\label{sec:evalErrors}There are two kinds of error in \texttt{numerica}:
those in the underlying \LaTeX{} which are reported in the \LaTeX{}
log, shown on the terminal, and generally halt compilation, and specifically
\texttt{numerica}-related errors which do not halt compilation and
produce messages displayed in the pdf where one would expect the result
of the calculation to be. The original reason for doing things this
way was to enable \texttt{numerica} to be used effectively with the
instant preview facility of the document processor \LyX . More philosophically,
one might view such errors as similar to grammatical errors or spelling
mistakes in text. It is not clear that they should halt compilation.
Hence strictly \texttt{numerica}-related errors leave brief messages
in the pdf at the offending places.

Before discussing specific error messages, note that there is a debug
facility (of a sort) discussed below in §\ref{subsec:settingsDebug}. 

Error messages are in two parts: a \emph{what} part and a \emph{where}
part.

\subsection{Mismatched brackets}

\label{subsec:errorsMismatched-brackets}An unmatched left parenthesis
or other left bracket (in this case a missing right parenthesis) usually
results in a \texttt{numerica} error:
\begin{centred}
\verb`$\eval{\sin(\pi/(1+x)}[x=1]$` $\Longrightarrow$ $\eval{\sin(\pi/(1+x)}[x=1]$
\end{centred}
For the same error in the vv-list, the what-part remains unchanged
but the where-part is altered:
\begin{centred}
\verb`$\eval{ 1+y }[x=1,y=\sin(\pi/(1+x)]$` $\Longrightarrow$ $\eval{ 1+y }[y=\sin(\pi/(1+x),x=1]$
\end{centred}
The \emph{what} message is the same; the \emph{where} is different.

An unmatched right parenthesis or other right bracket (in this case
a missing \emph{left} parenthesis) usually results in a similar \texttt{numerica}
error:
\begin{centred}
\verb`$\eval{2((x+y)/(y+z)))^2}[x=1,y=2,z=3]$` $\Longrightarrow$
\eval{2((x+y)/(y+z)))^{2}}[x=1,y=2,z=3]
\end{centred}
But note that an unmatched modifier like \verb`\left` or \verb`\right`
is a \LaTeX{} error and is caught by \LaTeX{} before \texttt{numerica}
can respond and so results in a terminal and logfile message. 

\subsection{Unknown tokens}

An `Unknown token' message can arise in a number of ways. If an
expression involves a number of variables, some of which depend on
others, their order in the vv-list matters: 
\noindent \begin{center}
\verb`$\eval{\tfrac12 vt}[t=2,v=gt,g=9.8]$` $\Longrightarrow$ \eval{\tfrac{1}{2}vt}[t=2,v=gt,g=9.8] 
\par\end{center}

The vv-list is evaluated from the \emph{right} so that in this example
the variable \texttt{v} depends on a quantity \texttt{t} that is not
yet defined. Hence the message. The remedy is to move \texttt{t} to
the right of \texttt{v} in the vv-list.

Similarly, if we use a variable in the formula that has not been assigned
a value in the vv-list, we again get the `Unknown token' message,
but this time the location is the formula:
\begin{centred}
\verb`$\eval{\pi r^2h}[r=3]$` $\Longrightarrow$ \eval{\pi r^{2}h}[r=3]
\end{centred}
The remedy obviously is to assign a value to \texttt{h} in the vv-list\texttt{.}

The same message will result if a mathematical operation or function
is used that has not been implemented in \texttt{numerica}:
\begin{centred}
\verb`$\eval{u \bmod v }[v=7,u=3]$` $\Longrightarrow$ \eval{u\bmod v}[v=7,u=3] 
\end{centred}
A missing comma in the vv-list will generally result in an unknown
token message:
\begin{centred}
\verb`$\eval{axy}[a=3 y=2,x=1]$` $\Longrightarrow$ \eval{axy}[a=3y=2,x=1] 
\end{centred}
Because of the missing comma, \texttt{numerica} assumes \texttt{a}
has the `value' \texttt{3y=2} and has no knowledge of \texttt{y}
as a variable.

The presence of multi-token variables can also cause this error message
if the check for such variables is turned off; see §\ref{subsec:settingsMultitokSwitch}.

\subsection{Other vv-list errors}

Other errors that can occur with the vv-list are overlooked value
assignments to variables, or missing commas. For the first, it is
essential that we do actually give a value to all variables occuring
in the vv-list:
\begin{centred}
\verb`$\eval{axy}[a=3,y=,x=1]$` $\Longrightarrow$ \eval{axy}[a=3,y=,x=1] 
\end{centred}
The remedy is obvious – assign a value to \texttt{$y$}. 

\emph{Extra} commas in the vv-list should cause no problems:
\begin{centred}
\verb`$\eval{axy}[,a=3,,y=2,x=1,]$` $\Longrightarrow$ $\eval{axy}[,a=3,,y=2,x=1,]$ 
\end{centred}

\subsection{Inverse powers of negative numbers}

\label{subsec:errorsInverse-powers}Inverse integer powers of positive
numbers should always be possible, but raising a \emph{negative} number
to an inverse power generates an error even when, mathematically,
it should not: 
\begin{centred}
\verb`\eval{$ (-125)^{1/3} $}` $\Longrightarrow$ \eval{$ (-125)^{1/3} $}
\end{centred}
This is a feature of floating point arithmetic. When a number is raised
to a rational power, say $p/q$ where $p$ and $q$ are non-zero integers,
then the result is the $p$-th power of the $q$-th root of the number.
Can a $q$-th root be taken? If our floating point system used (for
ease of illustration) only $4$ significant digits, $p/q=1/3$ would
be the fraction $3333/10^{4}$, an odd numerator over an even denominator.
But a negative number does not possess an even ($10^{4}$-th) root. 

\subsection{Integer argument errors}

\label{subsec:errorsIntegerArgs}Some functions require integer arguments
– factorials, the second argument of a binomial coefficient, and (in
\texttt{numerica}) $n$-th roots using the optional argument of \texttt{\textbackslash sqrt};
also summation and product variables. If integers are explicitly entered
for these arguments there is no problem, but if the value of the argument
is the result of a calculation, rounding errors require thinking about.
What accumulation of rounding errors is \emph{too} much so that the
result of the calculation \emph{cannot} be considered an integer?
\texttt{numerica} is generous: in the default setup, if a calculation
rounds to an integer at rounding value $14$ the result of the calculation
is considered an integer (obviously, the integer resulting from the
rounding). Since \texttt{l3fp} works to $16$ significant figures
that gives ample room for rounding errors to `get lost in' and be
ignored, while still ruling out such things as (recall the example
in §\ref{subsec:evalBoolean-output}),
\begin{centred}
\verb`\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]` $\Longrightarrow$
\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]
\end{centred}
where $N$ differs from $81$ not until the seventh decimal place. 

The default rounding value of $14$ for `int-ifying' calculations
can be changed: see §\ref{subsec:defaultsIntifyingRounding}.

\subsection{Comparison errors}

Should a user try to make a comparison using a combination like \verb`>=`
rather than \verb`\geq`, \texttt{numerica} admonishes like this: 
\begin{centred}
\verb`$\eval{ \pi^e >= e^\pi }[?]$` $\Longrightarrow$ $\eval{ \pi^e >= e^\pi }[?]$
\end{centred}
(The relation is false by the way.) 

\subsection{Invalid base for \texttt{\textbackslash log}}

ISO recommends using \verb`\log` only with a subscripted base specified.
Otherwise how is one to know whether the base is \verb`e` or $10$
or $2$ or whatever? Nonetheless \texttt{numerica} assumes that when
\verb`\log` is used unsubscripted, the base is 10. Suppose you want
to make $12$ the base, but forget to put braces around the $12$:
\begin{centred}
\verb`$\eval{ \log_12 1728 }$` $\Longrightarrow$ $\eval{ \log_12 1728 }$
\end{centred}
Here, \texttt{numerica} has taken \texttt{1} as the base (and $21728$
as the argument) of the logarithm and responds accordingly.

\subsection{\texttt{l3fp} errors}

Some errors arising at the \texttt{l3fp} level are trapped and a message
displayed. 
\begin{itemize}
\item Dividing by zero
\end{itemize}
\begin{centred}
\verb`$\eval{1/\sin x}[x=0]$` $\Longrightarrow$ \eval{1/\sin x}[x=0] 
\end{centred}
Note however that \verb`$\eval{1/\sin x}[x=\pi]$` $\Longrightarrow\,\eval{1/\sin x}[x=\pi]$,
because of rounding errors in distant decimal places. No doubt this
is true for other functions as well.
\begin{itemize}
\item Invalid operation
\end{itemize}
\begin{centred}
\verb`$\eval{\arccos x}[x=2]$` $\Longrightarrow$ \eval{\arccos x}[x=2]
\end{centred}
In this case the inverse cosine has been fed a value of $x$ outside
its domain of definition, the interval $[-1,1]$. Trying to evaluate
an expression that resolves to $0/0$ also produces this message:
\begin{centred}
\verb`$\eval{\frac{1-y}{x-2}}[x=2,y=1]$`$\Longrightarrow$ \eval{\frac{1-y}{x-2}}[x=2,y=1]
\end{centred}
\begin{itemize}
\item Overflow/underflow
\end{itemize}
The factorial (discussed in §\ref{subsec:evalFactorialsBinom}) provides
an example of overflow:
\begin{centred}
\verb`$\eval{3249!}$`\texttt{ }$\Longrightarrow$ \eval{3249!} 
\end{centred}
This is hardly surprising since
\begin{centred}
\verb`$\eval{3248!}[x]$` $\Longrightarrow$ $\eval{3248!}[x]$.
\end{centred}
There is a limit on the size of exponents that \texttt{l3fp} can handle.
A number in the form $a\times10^{b}$ must have $-10001\le b<10000$.
If this is not the case an overflow or underflow condition occurs.
As the examples show, an overflow condition generates a \texttt{numerica}
error. For underflow, where the number is closer to $0$ than $10^{-10001}$,
\texttt{l3fp} assigns a zero value to the quantity. \texttt{numerica}
accepts the zero value.

\chapter{Settings}

\label{chap:Settings}A calculation is effected against a background
of default values for various quantities. For a particular calculation,
these values may not be appropriate; or you may have different preferences.
The way to change settings for a particular calculation is through
the settings option of \verb`\nmcEvaluate` discussed next. The way
to change a \emph{default} setting is by creating a configuration
file \texttt{numerica.cfg} discussed in §\ref{sec:settingsDefaults}.

\section{Settings option}

\label{sec:settingsOption}The second argument of the \verb`\nmcEvaluate`
command is the settings option, delimited by square brackets. This
option is a \emph{key=value} list, hence comma-separated. \emph{Key=value
}lists tend to be wordy. For back-of-envelope calculations one wants
to be able to `dash off' the calculation, hence the short, cryptic
nsture of the keys. Most settings are generic, applicable not only
to \verb`\nmcEvaluate` but also to other commands that are available
if \texttt{numerica} is loaded with the \texttt{plus} option; see
the associated document \texttt{numerica-plus.pdf}. 

\subsection{\textquoteleft Debug\textquoteright{} facility}

\label{subsec:settingsDebug}It is rather grandiose to call this a
debug facility, but if a calculation goes wrong or produces a surprising
result, \texttt{numerica} offers a means of examining some intermediate
stages on the way to the final result. To use the facility, enter 
\begin{lyxcode}
dbg~=~<integer>
\end{lyxcode}
into the settings option. (White space around the equals sign is optional.) 
\begin{itemize}
\item \texttt{dbg=0 }turns off the debug function, displays the result or
error message (this is the default);
\item \texttt{dbg=1 }equivalent to \texttt{dbg=2{*}3{*}5{*}7};
\end{itemize}
The `magic' integers are the following primes and their products:

\begin{table}[t]
\centering
\noindent \centering{}\caption{Settings options}
\noindent \begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
\midrule
{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. arg. parsing} & {\small\texttt{0}}\tabularnewline
{\small\texttt{o}} &  & {\small degree switch for trig. funcions} & \tabularnewline
{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
{*} &  & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
in }{\small\texttt{vvd}} & \tabularnewline
{\small\texttt{p}} & token(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}

\begin{itemize}
\item \texttt{dbg=2} displays the vv-list after multi-token variables have
been converted to their single token form, \texttt{\textbackslash\_nmca},
\texttt{\textbackslash\_nmcb}, etc.;
\item \texttt{dbg=3} displays the formula after multi-token variables have
been converted to their single token form;
\item \texttt{dbg=5} displays the stored variables and their evaluated values
(\texttt{dbg=2} lists the values as expressions; here they have been
evaluated); note that any saved values (Chapter~\ref{chap:Saving-and-retrieving})
that have been loaded will also feature in this list which might lead
to a messy display depending on the nature of those values;
\item \texttt{dbg=7} displays the formula after it has been fp-ified (but
before it has been fed to \texttt{l3fp} to evaluate);
\begin{itemize}
\item should the formula successfully evaluate, the result of the evaluation
is also displayed.
\end{itemize}
\end{itemize}
\vspace{-1ex}

\begin{wraptable}{o}{0.5\columnwidth}%
\centering{}\caption{Magic integers}\label{tab:settingsMagic-integers}
\begin{center}
\begin{tabular}{rrl}
\toprule 
integer &  & \multicolumn{1}{c}{factors}\tabularnewline
\midrule 
6 &  & 2,3\tabularnewline
10 &  & 2,5\tabularnewline
14 &  & 2,7\tabularnewline
15 &  & 3,5\tabularnewline
21 &  & 3,7\tabularnewline
30 &  & 2,3,5\tabularnewline
35 &  & 5,7\tabularnewline
42 &  & 2,3,7\tabularnewline
70 &  & 2,5,7\tabularnewline
105 &  & 3,5,7\tabularnewline
210 &  & 2,3,5,7\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}\end{wraptable}%
To display two or more of these elements simultaneously, use the product
of their debug numbers for the magic integer. This can be entered
either as the multiplied-out product, or as the `waiting to be evaluated'
product with asterisks (stars) between the factors. Thus \texttt{dbg=6}
or \verb`dbg=2*3` display both the vv-list and formula after multi-token
variables have been converted to single token form; \texttt{dbg=10}
or \verb`dbg=2*5` display both the vv-list after multi-token variables
have been converted to single token form and the recorded variables
with their evaluated values. And similarly for the other magic integers
listed. For other integers, if they are divisible by $2$ or $3$
or $5$ or $7$, they will display the corresponding component. Both
\texttt{dbg=210} and \verb`dbg=2*3*5*7` display all four elements,
but rather than remembering this product, it suffices to put \texttt{dbg=1}.
This is equivalent and displays all elements.

The debug option uses an \texttt{aligned }or \texttt{align{*}} environment
to display its wares, depending on the presence or absence of math
delimiters around the \texttt{\textbackslash eval} command. In the
next example I have used multi-token variables to illustrate the different
elements in the debug display, and a chain of comparisons to show
how \texttt{numerica} treats these (§\ref{subsec:evalBoolean-output}).
\begin{verbatim}
    \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
      [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
\end{verbatim}
$\Longrightarrow$     \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
      [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']

Note that the four elements are displayed in temporal order: first
comes the vv-list after conversion of multi-token to single-token
variables, then the formula in the single-token variables; these are
created essentially at the same time. The vv-list is presented in
left-to-right order because that is the direction of evaluation \emph{internally}
in \texttt{numerica}. Next the stored values of the variables are
displayed. These are the values \emph{after }evaluation. The fourth
element both in the display and chronologically is the fp-ified formula;
this is often a thicket of parentheses. The final element of the display
and chronologically is the result of evaluating the formula. This
is displayed only if $7$ is a factor of the \texttt{dbg} integer,
and there is no error. 

When interpreting the fp-form, one should be aware of differences
between \texttt{numerica} and \texttt{l3fp}. In particular be aware
that in \texttt{l3fp} function calls bind most tightly so that, for
example, \verb`sin 2pi` evaluates not to zero but to $(\sin2)\times\pi$
and \verb`sin x^2` evaluates to $(\sin x)^{2}$. This should not
be of any concern to the user except as here in debug mode when interpreting
fp-forms. 

Finally, note that those mathematical operations that have no direct
representation in \texttt{l3fp} contribute only their value to the
fp-form. This applies to sums and products, double factorials and
partly to binomial coefficients as illustrated in the followng (ridiculous)
example:
\begin{verbatim}
    \eval[dbg=1]{\[ 
      \sum_{n=1}^5 n + \binom{10}{m}
        + \prod_{n=2}^5 (1-1/n) + m!! \][m=6]
\end{verbatim}
$\Longrightarrow$     \eval[dbg=1]{\[ 
      \sum_{n=1}^5 n+\binom{10}{m}
        +\prod_{n=2}^5(1-1/n) +m!! \]}[m=6]

\noindent The various contributions to the overall result are displayed
simply as numbers because \texttt{l3fp} does not (at least as yet)
handle these elements natively.

\subsubsection{Negative \texttt{dbg} values}

Negative \texttt{dbg} values are possible: \texttt{dbg=-2}, \texttt{dbg=-3},
etc. (and \texttt{dbg=-1} meaning \texttt{dbg=-210}) have exactly
the same effects as the corresponding positive values except for some
details of display. The display for positive \texttt{dbg} values is
the one evident in the examples above. Lines wrap, the left margin
is not indented and the display occupies the page width. For negative
\texttt{dbg} values, lines do not wrap, the left margin is indented
and the display occupies the text width. An example is presented in
§\ref{subsec:nestDebugging} below where the display for a nested
\verb`\eval` is significantly improved with a negative \texttt{dbg}
value.

\subsection{Reuse setting}

This setting determines whether the entire display or only the numerical
result is saved to file with the \verb`\nmcReuse` command. See below,
Chapter~\ref{chap:Saving-and-retrieving}, §\ref{subsec:reuse=00005Ceval's-reuse-setting}.

\subsection{Inputting numbers in scientific notation}

\label{subsec:settingsInputtingSciNotation}\emph{Outputting} numbers
in scientific notation is controlled by the final trailing argument
of the \texttt{\textbackslash eval} command. That is turned off by
default and needs to be explicitly ordered. Similarly, \emph{inputting}
numbers in scientific notation is turned off by default and needs
to be explicitly ordered. To turn it on, write
\begin{lyxcode}
\textasciicircum ~=~<char>
\end{lyxcode}
\noindent in the settings option, where \texttt{<char>} is any single
character, usually \texttt{e} or \texttt{d} or their upper-casings,
but not restricted to them: \texttt{\textasciicircum =@} for instance
is perfectly possible, and has the advantage over \texttt{e} or \texttt{d}
that it doesn't conflict with the use of the character as a variable
or constant. 
\begin{centred}
\verb`$ \eval[^=@]{ 1.23@-1 } $` $\Longrightarrow$ $ \eval[^=@]{ 1.23@-1 } $.
\end{centred}
With letters for the exponent mark – say \verb`d` or \verb`e` –
the problem is interpreting forms like \texttt{8d-3} or \texttt{2e-1}.
Does such a form denote a number in scientific notation or an algebraic
expression? In \texttt{numerica}, if the settings option shows \texttt{\textasciicircum =d},
then a form like \texttt{8d-3} is treated as a number in scientific
notation. Similarly for \texttt{e} or any other letter used as the
exponent marker for the input of scientific numbers. (But only one
character can be so used at a time.) Note that the number \emph{must}
start with a digit: \verb`e-1` for instance does not and will be
treated as an algebraic expression involving the exponential constant
(unless \verb`e` is assigned a different value in the vv-list).
\begin{centred}
\verb`$ \eval[^=e]{ x+e-1 }[x=2e-1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=2e-1] $. 
\end{centred}
The problem here is that \verb`2e-1` is treated as a number in scientific
notation but displays in the vv-list as if it were an algebraic expression.
The solution is to put \verb`2e-1` into an \verb`\mbox` or \verb`\text`
command in the vv-list:
\begin{centred}
\verb`$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $` $\Longrightarrow$
$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $. 
\end{centred}
If you use a particular character as the exponent marker for inputting
numbers in scientific notation, it is good practice \emph{not} to
use that character as a variable, not because it will cause an error
but because it makes expressions harder to read. 

\subsection{Multi-token variables}

\label{subsec:settingsMultitokSwitch}Variables need not consist of
a single character or token (like $x$ or $\alpha$). Multi-token
symbols like $x'$ or $t_{i}$ or $var$ are perfectly acceptable.
For its internal operations, \texttt{numerica} converts such multi-token
names to single tokens (as discussed in §\ref{subsec:evalVariableNames}).
This conversion takes time. Even if there are no multi-token variables
used at all, \texttt{numerica} still needs to check that that is so.
There is a setting that allows a user to turn off or turn on the check
for such variables by entering
\begin{lyxcode}
xx~=~<integer>
\end{lyxcode}
into the settings option. If \texttt{<integer>} is \texttt{0}, the
check for (and conversion of) multi-token variables is turned off;
if \texttt{<integer>} is \texttt{1} (or any other\emph{ non-zero}
integer), the check, and conversion if needed, goes ahead. By default,
checking for multi-token variables and converting them if found is
turned \emph{on}. (The name for the key, \texttt{xx}, is chosen because
\texttt{x} is the most familiar variable of all, introduced in elementary
algebra, and doubling it like this suggests multi-token-ness.)

If checking is turned off when a multi-token variable is present,
an error results. We don't need to enter \texttt{xx=1} in the first
of the following examples because the check for multi-token variables
is on by default. Explicitly turning it off in the second produces
an error.
\begin{centred}
\verb`\eval{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval{$ x_0^{\,2} $}[x_0=5],\medskip{}
\verb`\eval[xx=0]{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval[xx=0]{$ x_0^{\,2} $}[x_0=5]
\end{centred}

\subsection{Parsing arguments of trigonometric functions}

This setting allows a wider range of arguments to trigonometric  functions
to be used (think Fourier series) without needing to insert extra
parentheses in order for them to be read correctly by \verb`\eval`;
see §\ref{subsec:parseTrigFns}.

\subsection{Using degrees rather than radians}

\label{subsec:settingsDegrees}You may find it more convenient to
use degrees rather than radians with trigonometric functions.This
can be switched on simply by entering a lowercase \texttt{o} in the
settings option. (The author's fond hope is that the charitable eye
might see a degree symbol in the  \texttt{o}.) Thus
\begin{centred}
\verb`\eval[o]{$ \sin 30 $}` $\Longrightarrow$ \eval[o]{$ \sin 30 $},

\verb`\eval[o]{$ \arcsin 0.5 $}` $\Longrightarrow$ \eval[o]{$ \arcsin 0.5 $}.
\end{centred}

\subsection{Specifying a logarithm base}

\label{subsec:settingsLogBase}If you wish to use \verb`\log` without
a subscripted base in a particular calculation, then add an entry
like 
\begin{lyxcode}
log~=~<positive~number${}\,\mathtt{\ne1}$>
\end{lyxcode}
to the settings option of the \verb`\eval` command. The \verb`<positive number>`
does not need to be an integer. It could be \verb`e` (if you object
to writing \verb`\ln`) but is more likely to be $2$ or another small
integer; $10$ is the default. If you want to use this changed base
not for one but most calculations, then add an entry with your choice
of base to a configuration file; see §\ref{sec:settingsDefaults}.

\subsection{Calculation mode}

\label{subsec:settingsRecalcMode}A variable may change in the course
of a calculation. This is certainly true of sums and products. If
a parameter in the vv-list depends on the variable then that parameter
will need to be recalculated, perhaps repeatedly, in the course of
a calculation. By entering \label{subsec:settingsCalculationMode}
\begin{lyxcode}
vvmode~=~<integer>
\end{lyxcode}
in the settings option it is possible to turn on or off the ability
to repeatedly evaluate the vv-list; \verb`<integer>` here takes two
possible values, \texttt{0} or \texttt{1}. \texttt{vvmode=0} means
the vv-list is evaluated once at the start of the calculation; \texttt{vvmode=1}
means the vv-list is recalculated every time the relevant variable
changes.

For example, it may be desirable to place the summand, or some part
of it, in the vv-list. Since the summation variable obviously changes
during the course of the calculation, we need to enter \texttt{vvmode=1}
in the settings option. Repeating an earlier sum (the seting \verb`p=.`
is discussed in §\ref{subsec:settingsPunctuation}) \medskip{}
\begin{verbatim}
    \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
      [N=100,f(k)=1/k^3,{k}=1][4] 
\end{verbatim}
$\Longrightarrow$      \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
        [N=100,f(k)=1/k^3,{k}=1][4]

As you can see, the summand \texttt{f(k)} has been given explicit
form in the vv-list – equated to \texttt{1/k\textasciicircum 3}.
That means we need to give a preceding value to \texttt{k} in the
vv-list; hence the rightmost entry. But we don't want \texttt{k=1}
appearing in the final display, so we wrap \texttt{k} in braces (see
§\ref{subsec:evalVvSuppressVars}). Since the value \texttt{k=1} applies
only to the first term in the sum, to ensure it is not used for all
terms, we enter \texttt{vvmode=1} in the settings option. This turns
vv-recalculation mode on and ensures \texttt{k=1} is overwritten by
\texttt{k=2}, \texttt{k=3} and so on, and the vv-list recalculated
each time. The final result is the same as before, although recalculating
the vv-list at each step is a more resource-hungry process. The difference
may not be marked for this example; with more complicated expressions
it noticeably takes longer.

Because it is necessary to activate this switch when using \emph{implicit}
notations – like $f(k)$ in the example – rather than the explicit
form of the function in the main argumet it seems natural to call
\texttt{vvmode=1} \emph{implicit }mode and \texttt{vvmode=0} (the
default) \emph{explicit }mode. Most calculations are explicit mode
– the vv-list is evaluated only once.\emph{ }

\subsection{Changing the vv-list display format}

\label{subsec:settingsVvDisplayChangeLocal}In previous formulas with
variables the vv-list has been displayed following the result. It
is wrapped in parentheses following a comma followed by a space. These
formatting elements – comma, space, parentheses – can all be changed
with the settings option.

The default format specification is 
\begin{lyxcode}
\{,\}\textbackslash mskip~12mu~plus~6mu~minus~9mu(vv)
\end{lyxcode}
for a text-style display (an inline formula) and
\begin{lyxcode}
\{,\}\textbackslash mskip~36mu~minus~24mu(vv)
\end{lyxcode}
in a display-style context. The commas are wrapped in braces because
these are items in a comma-separated list. Both entries exhibit the
elements: punctuation (comma), preceding a variable space, preceding
the parenthesized vv-list (the \texttt{vv} placeholder). No full stop
is inserted after the closing parentheses because the \texttt{\textbackslash eval}
command may occur in the middle of a sentence (even in display style).
For inline use, the elasticity of the space becomes relevant when
\TeX{} is adjusting individual lines to fit sentences into paragraphs
and paragraphs into pages. The largest spacing that can be stretched
to is a quad, $18$~mu (mu~=~math unit), and the smallest that
can be shrunk to is a thin space, $3$~mu. In display style,\texttt{
}the largest spacing specified is the double quad, in line with the
recommendation in \emph{The \TeX{} Book}, Chapter~18, but this can
shrink to a single quad, for instance if the vv-list is heavily populated
with variables so that the evaluated result is pushed well to the
left by the vv-list. (But see below, §\ref{subsec:settings New-line-display}.)

If you want to change these defaults, enter in the settings option
\begin{lyxcode}
vvi~=~<new~specification>
\end{lyxcode}
to change the inline display and
\begin{lyxcode}
vvd~=~<new~specification>
\end{lyxcode}
to change the display-style display For example the settings
\begin{lyxcode}
vvi~=~\{,\}\textbackslash quad(vv)

vvd~=~\{,\}\textbackslash qquad(vv)
\end{lyxcode}
would give a comma (in braces since the settings option is a comma-separated
list) and a fixed space (of one or two quads) between the result and
the parenthesized vv-list. 

The vv-list itself in the display specification is represented by
the placeholder \texttt{vv}. If the \texttt{vv} is omitted from the
specification, then the vv-list will not appear at all:
\begin{centred}
\verb`\eval[vvi=?!]{$ \pi $}[\pi=3]` $\Longrightarrow$  \eval[vvi=?!]{$ \pi $}[\pi=3]
\end{centred}
More relevantly, it may well be the case that all variables in the
vv-list are suppressed (wrapped in braces). In that case the display
would look something like \verb`, ()`. To prevent this enter \verb`vvi=`
in the vv-list, in the inline case, or \verb`vvd=` in the display-style
case, i.e. enter an empty value. (Alternatively, use the star option
of the \verb`\eval` command.)

Another minor wrinkle occurs if you want to change parentheses around
the vv-list to square brackets. Because the settings option is a square-bracket
delimited argument, the square brackets in the specification will,
like commas, need to be hidden in braces, although you can get away
with braces around the whole spec.: 
\begin{lyxcode}
vvi=\{,\textbackslash mskip~12mu~plus~6mu~minus~9mu~{[}vv{]}\}
\end{lyxcode}

\subsection{Displaying the vv-list on a new line }

\label{subsec:settings New-line-display}Display of a long formula
with many variables, hence a full vv-list, may not fit comfortably
on a line. In an earlier example I used Brahmagupta's formula to calculate
the area of a triangle. It squeezed onto a line. I shall now use his
formula for the area of a cyclic quadrilateral:
\[
A=\sqrt{(s-a)(s-b)(s-c)(s-d)}.
\]
The extra side (quadrilateral as against triangle) means there is
a further variable to accommodate, not only in the formula but also
in the vv-list. In the following example, the cyclic quadrilateral
is formed by a 45-45-90 triangle of hypotenuse 2 joined along the
hypotenuse to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
Adding the areas of the two triangles, the area of the quadrilateral
is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
formula:
\begin{verbatim}
    \eval[p=.,vvd={,}\\(vv),*]
      {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
        [s=\tfrac12(a+b+c+d),
           a=\surd2,b=\surd2,c=\surd3,d=1] 
\end{verbatim}
$\Longrightarrow$     \eval[p=.,vvd={,}\\(vv),*]
      {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
        [s=\tfrac12(a+b+c+d), 
           a=\surd2,b=\surd2,c=\surd3,d=1] 

\noindent The values agree. The point to note here is the\texttt{
vvd=\{,\}\textbackslash\textbackslash (vv)} and the \texttt{{*}}
in the settings option. The \texttt{\textbackslash\textbackslash}
in a specification for \texttt{vvd} acts as a trigger for \texttt{numerica}
to replace whatever math delimiters are enclosed by the \verb`\eval`
command with a \verb`multline` environment. As you can see, the specification
inserts a comma after the formula and places the parenthesized vv-list
on a new line. The star \texttt{{*}} if present suppresses equation
numbering by turning the \verb`multline` into a \verb`multline*`
environment. 

Things to note in the use of\texttt{ \textbackslash\textbackslash}
in a \texttt{vvd} specification are that
\begin{itemize}
\item it applies only to the \texttt{vvd} specification, not the \texttt{vvi}
spec.;
\item it applies only when\emph{ }\verb`\eval`\emph{ wraps around }a math
environment of some kind; 
\item it has no effect when the \verb`\eval` command is used \emph{within}
a math environment when the presentation of the result is of the form
\emph{result, vv-list}. The formula is not displayed and so the pressure
on space is less and the `ordinary' vv-list specification is used.
\end{itemize}

\subsection{Punctuation}

\label{subsec:settingsPunctuation}The \verb`\eval` command can be
used within mathematical delimiters or it can be wrapped around mathematical
delimiters. The latter gives a \emph{formula=result} style of display
automatically, which is convenient. One doesn't need to write the
\emph{formula=} part of the expression, but it causes a problem when
\verb`\eval` wraps around a display-style or similar environment:
how to display a following punctuation mark? For an inline display
we can simply follow the \verb`\eval` command with the appropriate
punctuation, for instance: \verb`\eval{$ 1+1 $}.` $\Longrightarrow$
\eval{$ 1+1 $}. But with \verb`\[ \]` delimiters used \emph{within}
the \verb`\eval` command – \verb`\eval{\[ 1+1 \]}.` – the fullstop
slides off to the start of the next line, since it is beyond the closing
delimiter. We want it to display as if it were the last element \emph{before}
the closing delimiter.

Explicitly putting it there – \verb`\eval{\[ 1+1. \]}` – means the
punctuation mark becomes part of the formula. Potentially \texttt{numerica}
then needs to check not just for a fullstop but also other possible
punctuation marks like comma, semicolon, perhaps even exclamation
and question marks. All these marks have roles in mathematics or \texttt{l3fp}.
Including them in the formula means distinguishing their punctuation
role from their mathematical role and can only cause difficulties
(and code bloat).

Instead, \texttt{numerica} uses the setting 
\begin{lyxcode}
p~=~<char(s)>~
\end{lyxcode}
to place the \texttt{char(s)} after the result but within the environment
delimiters. The default punctuation mark is the comma so that simply
entering \texttt{p} will produce a comma in the appropriate place.
This saves having to write \texttt{p=\{,\}} as would otherwise be
required, since the settings option is a \emph{comma}-separated list.

Nor is one limited to a single punctuation mark: 
\begin{centred}
\verb`\eval[p=\ (but no 8!)]{\[ \frac{1}{81} \]}[9]` $\Longrightarrow$
\eval[p=\ \text{(but no 8!)}]{\[ \frac{1}{81} \]}[9]
\end{centred}

\section{Infinite sums and products}

\label{sec:settingsInfiniteSumsProds}There are ways of tweaking various
default settings to nudge infinite sums and products to a correct
limit. These tweaks are applied via the settings option of the \verb`\eval`
command. 

\begin{table}
\centering
\noindent \centering{}\caption{Settings for infinite sums \& products}\label{tab:settingsSumsProducts}
\noindent \begin{center}
{\small{}%
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
key & type & meaning & default\tabularnewline
\midrule 
\texttt{S+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
\texttt{S?} & $\text{int}\ge0$ & stopping criterion query terms for sums & \texttt{0}\tabularnewline
\texttt{P+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
\texttt{P?} & $\text{int}\ge0$ & stopping criterion query terms for products & \texttt{0}\tabularnewline
\bottomrule
\end{tabular}}
\par\end{center}
\end{table}
The normal convergence criterion used by \texttt{numerica} to determine
when to stop adding/multiplying terms in an infinite sum/product is
\emph{when the next term added/multiplied leaves the total unaltered
when rounded to 2 more digits than the specified rounding value.}
Suppose $T_{k}$ is the sum/product after the inclusion of $k$ terms,
and $r$ is the rounding value. Denote $T_{k}$ rounded to $r$ figures
by $\left(T_{k}\right)_{r}$. \emph{The infinite sum or product stops
at the $(k+1)$-th term (and the value is attained at the $k$-th
term) when }$\left(T_{k+1}\right)_{r+2}=\left(T_{k}\right)_{r+2}$.
The hope is that if this is true at rounding value $r+2$ then at
rounding value $r$ the series or product will have attained a stable
value at that level of rounding.

For a series of monotonic terms converging quickly to a limit, this
stopping criterion works well, less so if convergence is slower, as
seen earlier with the infinite product for $\sinh1$. The criterion
can fail completely when terms behave in a non-monotonic manner. Distant
terms of a Fourier series, for example, may take zero values; the
criterion is necessarily satisfied but the series may still be far
from its limit. In a product the equivalent would be a distant term
taking unit value. Such series or products may also have initial `irregular'
terms including zero/unit terms. A summation/product might stop after
only one or two additions/multiplications if the criterion were applied
to them. 

To cope with these possibilities, \texttt{numerica} offers two settings
for sums, two for products, summarized in Table~\ref{tab:settingsSumsProducts}.
These are entered in the settings option of the \verb`\eval` command.
\begin{itemize}
\item \texttt{S+=<integer> }or \texttt{P+=<integer>} additional rounding
on top of the specified (or default) rounding for the calculation;
default = $2$
\begin{itemize}
\item the larger the additional \texttt{<integer>} is, the more likely that
sum or product has attained a stable value at the specified rounding
$r$ 
\end{itemize}
\item \texttt{S?=<integer${}\,\mathtt{\ge0}$> }or \texttt{P?=<integer${}\,\mathtt{\ge0}$>}
the number of final terms to query after the stopping criterion has
been achieved to confirm that it is not an `accident' of particular
values; default = $0$
\begin{itemize}
\item a final few terms to be summed/multiplied and the rounded result after
each such operation to be compared with the rounded result at the
time the stopping criterion was achieved. Suppose the additional rounding
(\texttt{S+} or \texttt{P+}) is $n$ on top of the specified rounding
$r$ and let the number of final checking terms be $m$. Suppose $T_{k_{0}}$
is the first term at which the stopping criterion is achieved: $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1}\right)_{r+n}$.
What we require of the final query terms is that $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1+j}\right)_{r+n}$
for $j=0,1,\ldots,m$.
\end{itemize}
\end{itemize}
Previously we found that the infinite product for $\sinh1$ with the
default settings gave the wrong value, $0.174$, deficient by $1$
in the last digit. We now have the means to tweak the stopping criterion
by increasing the additional rounding: 
\begin{verbatim}
    \eval[p,P+=3]{\[
      \prod_{k=1}^{\infty}
      \biggl(\frac{x^2}{k^2\pi^2} +1\biggr) 
    \]}[x=1][3] \nmcInfo{prod}.
\end{verbatim}
\noindent $\Longrightarrow$ \noindent \eval[p,P+=3]{\[
  \prod_{k=1}^{\infty}
  \biggl(\frac{x^2}{k^2\pi^2} +1\biggr) 
\]}[x=1][3]  \nmcInfo{prod}.

\noindent To obtain that last item of information (350 factors), I've
anticipated a little and used the command \verb`\nmcInfo` with the
argument \verb`prod`; see Chapter~\ref{sec:info}. The product now
produces the correct three-figure value, but it takes $350$ factors
to do so. 

Knowing how many terms or factors have been needed helps assess how
trustworthy the result from an infinite sum or product is. For example,
for the exponential series,
\begin{verbatim}
    \eval[p]{\[
      \sum_{k=0}^\infty \frac1{k!} 
    \]}[9] \nmcInfo{sum}.
\end{verbatim}
$\Longrightarrow$ \eval[p]{\[\sum_{k=0}^\infty \frac1{k!} \]}[9] \nmcInfo{sum}. 

To $9$ places of decimals, using the default value \texttt{S+=2},
the exponential series arrives at the right sum after only $15$ terms.
Convergence is rapid. We can trust this result (and it is in fact
the correct nine-figure value). By contrast, if we didn't know the
value of $\sinh1$ beforehand, noting the number of factors required
would make us justly cautious about accepting the result of the infinite
product calculation. 

One way to gain confidence in a result is to choose a possibly unrealistic
rounding value – say, the default $6$ for the infinite product \-\-
then use \emph{negative} values for the extra rounding, \texttt{S+=-5},
\texttt{S+=-4}, \ldots{} , so that the stopping criterion applies at
rounding values $s$ of $6+(-5)=1$, $6+(-4)=2$, and so on, but the
result is always presented to $6$ decimal places. One can then see
how the $6$-figure results behave relative to the number of terms
it takes to meet the stopping criterion. A little experimenting shows
that for our infinite product for $\sinh1$ the number of factors
$N_{s}$ at a stopping rounding value $s$ increases in geometric
proportion with a scale factor of about $3$: $N_{s+1}/N_{s}\approx3$.
For the exponential series on the other hand $N_{s}=4+s$, the number
of terms increasing in direct proportion to the stopping rounding
value.

A similar calculation for the sum of inverse fourth powers of the
integers $\zeta(4)=\sum_{n=1}^{\infty}\frac{1}{n^{4}}$, inverse third
powers, $\zeta(3)$, and inverse squares, $\zeta(2)$, using \verb`\nmcInfo`
to find how many terms are required at each stopping rounding value,
shows that at least over the rounding value range $1$ to $8$, for
inverse fourth powers $N_{s+1}/N_{s}\approx1.7$, for inverse third
powers $N_{s+1}/N_{s}\gtrsim2$ and for inverse squares $N_{s+1}/N_{s}\approx3$.
All are geometric rather than arithmetic progressions, but for inverse
fourth powers the scale factor ($\approx1.7$) is sufficiently small
that for these low values of $s$ the number of terms required does
not grow too quickly. It is a standard result (Euler) that the series
sums to $\pi^{4}/90$: \verb`$ \eval{ \pi^4/90 } $` $\Longrightarrow$
$ \eval{ \pi^4/90 } $ to six places, and indeed, with the default
\texttt{S+=2}, 
\begin{centred}
\verb`\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}` $\Longrightarrow$
\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}
\end{centred}
there is complete agreement. 

For inverse third powers, the number of terms required to reach the
stopping criterion grows rapidly for rounding values from $7$ onwards
($2^{7}=128$, $2^{8}=256$, \ldots ). This suggests trying for a
five-figure result (with the default setting \texttt{S+=2} the stopping
rounding value is $7$). Doing this gives a result $1.20205$ to five
decimal places. \emph{HMF} Table 23.3 has this quantity tabulated
to $20$ places and shows our result is too small by $1$ in the final
figure. 

For inverse second powers, the number of terms required to reach the
stopping criterion increase even more quickly: $3^{4}=81$, $3^{5}=243$,
and so on. A three figure answer (with the default setting \texttt{S+=2}
the stopping rounding value is $5$) seems the best we can hope for.
Doing the evaluation gives $1.642$ whereas we know that $\zeta(2)=\pi^{2}/6$
(Euler's famous result), evaluating to $1.645$. Even with \texttt{S+=3},
the sum is still too small, $1.644$ after 1007 terms. Increasing
the additional rounding to $4$, \texttt{S+=4}, does finally give
the correct three-figure result, $1.645$, but only after summing
$3180$ terms.

\subsection{Premature ending of infinite sums}

All the series considered so far have been monotonic. Trigonometric
series will generally not be so, nor even single-signed. 

Trigonometric sums are computationally intensive and so, for the following
example, I have specified a rounding value of 2. The series 
\[
\sum_{n=1}^{\infty}\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt
\]
is the Fourier series for the triangular wave function /\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash{} \ldots{}
of period 1, symmetric about the origin where it takes its maximum
value 1, crossing the axis at $t=0.25$ and descending to its minimum
$-1$ at $t=0.5$, before ascending to a second maximum at $t=1$
(and so on). In the interval $[0,0.5)$ the series should sum to $1-4t$.
The problem is that the summand $\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt$
vanishes both when $n$ is even and when $4nt$ is an odd integer.
If $t=0.1$ then $4nt$ is never an odd integer so the summand vanishes
only for $n$ even, every second term. We expect the result to be
$1-4\times0.1=0.6$.
\begin{verbatim}
    \eval[p]{\[
      \sum_{n=1}^{\infty}
        \frac{4}{n^{2}\pi^{2}}
        (1-\cos n\pi)\cos2\pi nt 
    \]}[t=0.1][2] \nmcInfo{sum}.
\end{verbatim}
$\Longrightarrow$ \eval[p]{\[
  \sum_{n=1}^{\infty}
    \frac{4}{n^{2}\pi^{2}}
    (1-\cos n\pi)\cos2\pi nt 
\]}[t=0.1][2]  \info{sum}.

\noindent Only one term? Of course – since the second term $n$ is
even; the term vanishes and the stopping criterion is satisfied. The
way around this problem is to query terms \emph{beyond} the one where
the stopping criterion is achieved, i.e., to set \texttt{S?} to a
nonzero value. We try \texttt{S?=1}: 
\begin{verbatim}
    \eval[p,S?=1]{\[
      \sum_{n=1}^{\infty}
        \frac{4}{n^{2}\pi^{2}}
        (1-\cos n\pi)\cos2\pi nt 
    \]}[t=0.1][2] \nmcInfo{sum}.
\end{verbatim}
$\Longrightarrow$ \eval[p,S?=1]{\[
  \sum_{n=1}^{\infty}
    \frac{4}{n^{2}\pi^{2}}
    (1-\cos n\pi)\cos2\pi nt 
\]}[t=0.1][2] \info{sum}.

Table~\ref{tab:settingsFinite-sums} lists the results of evaluating
the \emph{finite }sums from $n=1$ to $N$ for values of $N$ around
$65$. Since we have specified a rounding value of $2$ for the calculation,
the stopping criterion\emph{ }applies at a rounding value of $2$
more than that, $4$. Since $N=64$ is even, the summand for the $64$th
term is zero and the sum takes the same value as for $N=63$. The
$65$th term is the query term and the sum differs, so the summation
continues. The $66$th term vanishes, so the stopping criterion is
met. This time for the query term, the $67$th, the sum retains the
same $4$-figure value, and the summation stops. The result was attained
at the $65$th term.

\begin{wraptable}{o}{0.5\columnwidth}%
\centering{}\vspace{-4ex}
\caption{Finite sums}\label{tab:settingsFinite-sums}
\setlength{\abovetopsep}{1ex}%
\begin{tabular}{cc}
\toprule 
$N$ & $\Sigma$\tabularnewline
\midrule
$63$ & $0.6001$\tabularnewline
$64$ & $0.6001$\tabularnewline
$65$ & $0.5999$\tabularnewline
$66$ & $0.5999$\tabularnewline
$67$ & $0.5999$\tabularnewline
\bottomrule
\end{tabular}\end{wraptable}%
Should we be confident in the result? Increase the number of query
terms to $3$ (there is no point in increasing \texttt{S?} to $2$
because of the vanishing of the even terms), the sum stops after $113$
terms, with the same $0.6$ result. Indeed, increasing \texttt{S?}
to $5,7,\ldots$ makes no difference. It still takes $113$ terms
to reach the stable two-figure result $0.6$. 

For a final example, consider the error function 
\[
\erf z=\dfrac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt
\]
which can also be rendered as an infinite sum (\emph{HMF }7.1.5):
\[
\erf z=\sum_{n=0}^{\infty}(-1)^{n}\frac{z^{2n+1}}{n!(2n+1)}.
\]
(\verb`\erf` expanding to \verb`erf` has been defined in the preamble
to this document using \verb`\DeclareMathOperator`.) We calculate
this sum for $z=2$ to $10$ places of decimals. Although this is
an alternating series, it is obvious that the summand never vanishes
when $z\ne0$ as here. Hence there seems no need to change the default
value \texttt{S?=0}.
\begin{verbatim}
    \eval[p]{\[ 
      \frac2{\sqrt{\pi}}
        \sum_{n=0}^\infty(-1)^n
          \frac{z^{2n+1}}{n!(2n+1)}   
    \]}[z=2][10*] \nmcInfo{sum}.
\end{verbatim}
$\Longrightarrow$ \eval[p]{\[ 
  \frac2{\sqrt{\pi}}
    \sum_{n=0}^\infty(-1)^n
      \frac{z^{2n+1}}{n!(2n+1)}   
\]}[z=2][10*] \nmcInfo{sum}.

According to \emph{HMF }Table 7.1, this calculated value of $\erf2$
is correct to all $10$ places. But beyond $z=2$ errors will begin
to interfere with the result. Note that $26$ terms means $n=26$
was the last value of $n$ for which the summand was evaluated. (The
sum stops at the $26$th term, $n=25$, but the next term $n=26$
needs to be calculated for the stopping criterion.) Fortuitously,
$2^{2\times26+1}=2^{53}$ is the greatest power of $2$ that can be
\emph{exactly} rendered to the $16$ significant figures that \texttt{l3fp}
uses. But $n!$ exceeds the $16$-significant figure limit of \texttt{l3fp}
when $n>21$, so despite the 10-figure result, errors have already
begun to occur in the denominator of the summand and accrue in the
sum when $z=2$. For larger $z$ values the errors can only get worse
and at some point will render the calculated value worthless at any
meaningful rounding value. For example, when $z=7$ the sum apparently
`evaluates' to over $929$ whereas we know that 
\[
\erf z<\dfrac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-t^{2}}dt=1.
\]


\subsection{Double sums or products}

Sums or products can be iterated. For instance, the exponential function
can be calculated this way:
\begin{centred}
\verb`\eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]`
$\Longrightarrow$ \eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
\end{centred}
\noindent which is \verb`\eval{$ e^2 $}` $\Longrightarrow\eval{\ensuremath{e^{2}}}$.

A second example is afforded by Euler's transformation of series (\emph{HMF}
3.6.27). To calculate $e^{-1}$ we use 
\begin{verbatim}
    \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]
      {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3] 
\end{verbatim}
$\Longrightarrow$ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]

(Note the placement of the information command as the value of the
punctuation key. This keeps it \emph{within }the \texttt{\textbackslash{[}
\textbackslash{]}} delimiters.) Following Euler, this series can
be transformed to the form
\begin{verbatim}
    \eval[p,S?=1]{\[
      \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}
      \sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} 
    \]}[3] \nmcInfo{sum}.
\end{verbatim}
$\Longrightarrow$ \eval[p,S?=1]{\[ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}\sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} \]}[3] \nmcInfo{sum}.

\noindent Note the setting \verb`S?=1`. Without it, the summation
stops after $1$ term, the $k=0$ term, because the $k=1$ term vanishes.
With \verb`S?=1` it takes $16$ terms of the \emph{outer }sum to
reach the stopping criterion. Since that sum starts at $0$, that
means that changing the upper limit from $\infty$ to $15$ should
give the same result – which it does – but it takes $\tfrac{1}{2}\times16\times17=136$
terms in total to get there, to be compared with the $9$ terms of
the earlier simpler sum, and the terms are more complicated. Obviously
such double sums are computationally intensive.

\section{Changing default values}

\label{sec:settingsDefaults}The settings option enables various settings
to be changed for an individual calculation. You may find yourself
wanting to make such changes sufficiently often that a change of default
value is a better plan than encumbering each calculation with a list
of settings.

\begin{table}[h]
\centering
\noindent \centering{}\caption{Default values, \texttt{\textbackslash eval} command}\label{tab:settingsDefaults}
\noindent \begin{center}
\begin{tabular}{ll}
\toprule 
{\small key} & {\small value}\tabularnewline
\midrule
{\small rounding} & {\small\texttt{6}}\tabularnewline
{\small pad} & {\small\texttt{0}}\tabularnewline
{\small output-sci-notation } & {\small\texttt{0}}\tabularnewline
{\small output-exponent-char} & {\small\texttt{e}}\tabularnewline
{\small\%} & \tabularnewline
{\small input-sci-notation } & {\small\texttt{0}}\tabularnewline
{\small input-exponent-char} & {\small\texttt{e}}\tabularnewline
{\small multitoken-variables } & {\small\texttt{1}}\tabularnewline
logarithm-base & {\small\texttt{10}}\tabularnewline
{\small vv-display} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
{\small vv-inline} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
{\small\%} & \tabularnewline
{\small intify-rounding} & {\small\texttt{14}}\tabularnewline
\% & \tabularnewline
{\small sum-extra-rounding} & {\small\texttt{2}}\tabularnewline
{\small sum-query-terms} & {\small\texttt{0}}\tabularnewline
{\small prod-extra-rounding} & {\small\texttt{2}}\tabularnewline
{\small prod-query-terms} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}
The way to do that is to create a \emph{configuration file }with the
name \texttt{numerica.cfg} in a text editor. Its entries, one per
line, are of the form \emph{key=value} followed by a comma, and for
clarity preferably one entry per line (although this is not essential).The
key names are noticeably more verbose than the corresponding keys
of the settings option.\emph{ }The possible keys are listed in Table~\ref{tab:settingsDefaults},
together with their current default values. 

Keys taking one of two possible values, \verb`0` (for \verb`false/off`)
or \verb`1` (for \verb`true/on`), are \verb`pad` (the result with
zeros), \verb`output-sci-notation`, \verb`input-sci-notation`, and
(check for) \verb`multitoken-variables`.

The table is divided into four parts. 
\begin{itemize}
\item The top four rows concern elements that can be changed for individual
calculations with the trailing optional argument of \verb`\eval`:
rounding, padding with zeros, and outputting in scientific notation;
see §\ref{subsec:evalRoundingEtc}. 
\begin{itemize}
\item Note that to output the result always in scientific notation requires
two settings, first setting \texttt{output-sci-notation} to \texttt{1},
and then choosing a character to act as the exponent marker. Because
\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
has made \texttt{e} its default. But this option is turned off by
default (hence the \texttt{0} against this key).
\end{itemize}
\item The next block of rows concern general elements that can be changed
for individual calculations with the settings option of \verb`\eval`;
see §\ref{sec:settingsOption}. Obviously the key names are more expansive
in the present context but the effect is the same. 
\begin{itemize}
\item But note that to input numbers in scientific notation requires two
settings, first setting \texttt{input-sci-notation} to \texttt{1},
and then choosing a character to act as the exponent marker. Because
\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
has made \texttt{e} its default. The option is turned off by default
(hence the \texttt{0} against this key). 
\end{itemize}
\item The third block is a single row specifying at what rounding value
a floating point should be considered an integer; see §\ref{subsec:defaultsIntifyingRounding}
below.
\item The last four rows concern default settings for infinite sums and
products. These correspond to the keys \texttt{S+}, \texttt{S?} and
\texttt{P+}, \texttt{P?} of the settings option that can be used to
tweak the behaviour of the stopping criterion for such sums or products;
see §\ref{sec:settingsInfiniteSumsProds}.
\end{itemize}
If you are dissatisfied with any of the default values listed, then
in a text editor create a new file called \texttt{numerica.cfg} and
assign \emph{your} values to the relevant keys. For instance, if you
find yourself working to $4$ figures, that rounding to $6$ is too
many, then make the entry \texttt{rounding=}4. If also you want results
always presented in proper scientific notation,\emph{ $d.d_{1}d_{2}d_{3}d_{4}\times10^{n}$},
then add a comma after \texttt{4} and enter on a new line (recommended
but not strictly necessary; the comma is the crucial thing), \texttt{output-sci-notation=1,}
(note the comma) and on another new line, \texttt{output-exponent-char=x}. 

Perhaps you also want a non-zero setting for the final query terms
for infinite sums and products. This makes sense if you are largely
dealing with non-monotonic series – like Fourier series. Even the
Euler transformation of the exponential series for $e^{-1}$ discussed
above required a non-zero \texttt{S?}. If you wish to make this change
then add a comma and on a new line add (for instance) \texttt{sum-query-terms
= 1,} and again on a new line, \texttt{prod-query-terms = }1. If this
is all you wish to change, then no comma is necessary after this final
entry. Your newly created file should look something like
\begin{lyxcode}
rounding~~~~~~~~~~~~~=~4,

output-sci-notation~~=~1,

output-exponent-char~=~x,

sum-query-terms~~~~~~=~1,

prod-query-terms~~~~~=~1
\end{lyxcode}
The white spacing may be different; white space is ignored by \texttt{numerica}
when reading the file. Using it to align the equals signs helps \emph{us}
read the file. Note that the last entry, because it is the last entry,
lacks a comma. Now save the file with the name \texttt{numerica.cfg}.
This file will be read by \texttt{numerica} near the end of its loading
process. These settings will be \texttt{numerica}'s defaults for the
relevant keys. 

\subsection{Location of \texttt{numerica.cfg}}

Save, yes, but where to? If the new settings are likely to apply only
to your current document, then the document's directory is a sensible
place to put it and \texttt{numerica} will certainly find it there
since it is part of \LaTeX 3 file handling that file searches are
not limited to the \TeX{} distribution (including your personal texmf
tree) but also include the current document directory. But what happens
when you start working on another document? Will you remember to copy
\texttt{numerica.cfg} to its new location? That is why your \emph{personal
texmf tree} is a better place. 

\subsubsection{Personal texmf tree? }

This is a directory for `waifs and strays' of the \TeX{} system that
are not included in the standard distributions like MiK\TeX{} or \TeX Live.
Here you place personal packages designed for your own particular
circumstances. These may include your own \TeX{} or \LaTeX{} package,
say \texttt{mypackage.sty}, achieving some small or singular effect
that doesn't warrant wider distribution on CTAN. Here you might place
configuration files for other packages with your preferences (unless
the package requires some specific location). Here you can put your
personal bibliography files. 

Your personal texmf tree is structured like the standard MiK\TeX{}
or \TeX Live hierarchy but placed in another location so that there
is no chance of its being overwritten when packages in MiK\TeX{} or
\TeX Live are updated. But these distributions need to be alerted
to its existence. 

For example, in the MiK\TeX{} console, click on \textsf{Settings},
and then on the \textsf{Directories} tab of the resulting dialog.
Here you get to add your personal texmf hierarchy to the list of paths
that MiK\TeX{} searches, by clicking on the \textsf{+} button, browsing
to your texmf folder and selecting it. By using the up and down arrow
keys that the MiK\TeX{} console provides, ensure that it lies \emph{above
}the the entry for the main MiK\TeX{} tree. That way, files in your
personal texmf tree will be found first and loaded. Now go to the
\textsf{Tasks} menu and click on \textsf{Refresh the filename database}.
This will let MiK\TeX{} know what is held in your personal texmf tree.
Files there can then be used like standard \LaTeX{} packages.

\subsection{Rounding in \textquoteleft int-ifying\textquoteright{} calculations}

\label{subsec:defaultsIntifyingRounding}Factorials, binomial coefficients,
summation and product variables, and (in \texttt{numerica}) $n$-th
roots from the \texttt{\textbackslash sqrt} command, all require
integer arguments. These integers may indeed be entered explicitly
as integers, but they can also be determined as the result of a calculation.
Rounding errors may mean the result is not an exact integer. How much
leeway should be allowed before it is clear that the calculation did
not give an integer result? In the default setup, \texttt{numerica}
is generous. A number is considered an integer if it rounds to an
integer when the rounding value is $14$. Since \texttt{l3fp} works
to $16$ significant figures this provides more than enough `elbowroom'
for innocuous rounding errors to be accommodated. If a calculation
does not round to an integer at a rounding value of $14$ then it
seems reasonable to conclude that it has \emph{really }not given an
integer answer, not just that rounding errors have accumulated. If
you want to change this `int-ifying' value for a particular calculation,
then add a line to \texttt{numerica.cfg} like
\begin{lyxcode}
intify-rounding~=~<integer>
\end{lyxcode}
Since \texttt{l3fp} works to $16$ significant figures, values of
\texttt{integer} greater than $16$ are pointless. Generally int-ifying
rounding values will be less than but close to $16$ (although when
testing the code I used some ridiculous values like $3$ or $4$).
If other entries follow this one in the file, then conclude the line
with a comma.

\section{Parsing mathematical arguments}

\label{sec:Argument-parsing} A main aim of the \texttt{numerica}
package is to require minimal, preferably no, adjustment to the \LaTeX{}
form in which an expression is typeset in order to evaluate it. But
mathematicians do not follow codified rules of the kind programming
languages insist on when writing formulas – like parenthesizing the
arguments of functions, or inserting explicit multiplication signs
({*}) between juxtaposed terms. Hence the question of where the arguments
of mathematical functions end is acute. For a few functions \LaTeX{}
delimits the argument: think of \verb`\sqrt`, \verb`\frac`, \verb`\binom`;
also \verb`^`. But for functions like \verb`\sin` or \verb`\tanh`
or \verb`\ln`, unary functions, this is not so. Nor is it for sums
and products, and comparisons.

Before discussing the parsing rules for different groups of functions,
I discuss the means \texttt{numerica} provides to handle exceptions
to those rules, when one \emph{does} need to make some adjustment
to a formula.

\subsection{The cleave commands \texttt{\textbackslash q} and \texttt{\textbackslash Q}}

\label{subsec:parseCleave}The word \emph{cleave} has two opposed
meanings: to adhere or cling to, and to split or sever. \texttt{numerica}
defines two commands, \verb`\q` and \verb`\Q` to achieve these opposite
effects. When a mathematical argument is being parsed, the \verb`\q`
command joins the next token to the argument (\emph{cleaves to});
the \verb`\Q` command severs the next token from the argument (\emph{cleaves
apart}). Neither command is added to the argument nor leaves a visible
trace in the output. 

Thus, without \verb`\q`, 
\begin{centred}
\verb`\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234],
\end{centred}
which is $(\sin\tfrac{7}{2})\times\pi$. With \verb`\q` between the
bracketed factors,
\begin{centred}
\verb`\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234],
\end{centred}
which is $\sin(\tfrac{7}{2}\pi)$. Similarly, without \verb`\q`,
\begin{centred}
\verb`\eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]`
$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]
\end{centred}
which is $(\cos\pi)\times3\times(1+\tfrac{1}{2}\times2)$. With \verb`\q`
used twice, once after the fraction and once before the left parenthesis,
\begin{centred}
\verb`\eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]`
$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]
\end{centred}
which is $\cos(\pi\times3\times2)$.

It should be noted that for \emph{trigonometric }functions, because
of their use in Fourier series especially, there is another way of
handling arguments with parentheses (and fractions). This is discussed
in §\ref{subsec:parseTrigFns} below.

For the \verb`\Q` command which splits an argument we have, without
it,
\begin{centred}
\verb`\eval{$ 1/2e $}` $\Longrightarrow$ \eval{$ 1/2e $}, 
\end{centred}
which is the reciprocal of $2e$, whereas with the \verb`\Q` command
inserted before \verb`e`,
\begin{centred}
\verb`\eval{$ 1/2\Q e $}` $\Longrightarrow$ \eval{$ 1/2\Q e $},
\end{centred}
which is a half of $e$. Of course, the meaning in this example would
be clearer if $1/2$ were parenthesized or presented as a \verb`\tfrac`.

\subsubsection{Mnemonic}

As mnemonic, best seen in sans serif for the Latin Modern fonts used
in this document, think of the letter \textsf{q} as a circle \emph{clinging
}to a vertical descender; think of the letter \textsf{Q} as a circle
\emph{cut} by the diagonal stroke.

\subsection{Parsing groups}

The arguments of different groups of functions are handled in different
ways. The criterion used for deciding when an argument ends for one
group will not be that used for the others. Table §\ref{subsec:defaultsIntifyingRounding}
lists the different groups that \texttt{numerica} takes account of.
At the top are functions or operations that have the smallest reach
when determining where their arguments end; at the bottom are operations
that have the greatest reach. The denominator of a slash fraction
is treated as a unary function and is assigned to group II. By default
trigonometric functions are treated the same as other unary functions
but there is a setting which enables the direct (rather than inverse)
trigonometric functions to accept a wider range of arguments, as occurs
in Fourier series. Hence they are separated into their own group.

\begin{wraptable}{o}{0.5\columnwidth}%
\noindent \centering{}\caption{Parsing groups}\label{tab:settingsParsing-groups}
{\ttfamily\noindent \begin{center}
{\ttfamily{}%
\begin{tabular}{ll}
\toprule 
{\small\textrm{group}} & {\small\textrm{function/operation}}\tabularnewline
\midrule
{\small\textrm{I}} & {\small\textrm{surd, logical Not}}\tabularnewline
{\small\textrm{II}} & {\small\textrm{unary functions, /}}\tabularnewline
{\small\textrm{III}} & {\small\textrm{direct trig. functions}}\tabularnewline
{\small\textrm{IV}} & {\small\textrm{sums, products}}\tabularnewline
{\small\textrm{V}} & {\small\textrm{comparisons}}\tabularnewline
{\small\textrm{VI}} & {\small\textrm{logical And, logical Or}}\tabularnewline
\bottomrule
\end{tabular}}
\par\end{center}}\end{wraptable}%
 A formula is a sequence of tokens and brace groups. All parsing occurs
from the left, \LaTeX{} argument by \LaTeX{} argument, where \emph{argument}
means either a token (an N-type argument in \verb`expl3`-speak) or
a brace group (an n-type argument). To distinguish \LaTeX{} arguments
from mathematical arguments I shall when necessary refer to L-args
and M-args. A mathematical argument may end \emph{at} an L-arg, meaning
immediately before the L-arg, or end \emph{with} the L-arg, meaning
immediately after the L-arg. Ending or not will in general depend
on whether the argument is in \emph{first position} – the position
immediately following a function token like \verb`\sin` or \verb`\log`
– or in \emph{general position} – any later position (although for
trigonometric functions we will also need to consider \emph{second}
and even \emph{third }position). 

For counting position, we need to allow for formatting elements and
multi-token numbers – in both decimal and scientific formats. Formatting
elements do not change the position count. This applies to things
like thin spaces or phantoms (and their arguments) or modifiers like
\verb`\left` or \verb`\biggl`. Multi-token numbers (in decimal or
scientific formats) are treated as single items; they advance the
position count by exactly one. \LaTeX{} functions – like \verb`\frac`
– which take \LaTeX{} arguments again advance the position count only
by one. Mathematically, the fraction is viewed as a single unit.

I shall refer to a token or a token and its \LaTeX{} arguments – like
\verb`\frac` and its arguments – as an \emph{item}. Similarly, a
(possibly multi-token) number is an item. Also it will help to distinguish
tokens within brackets where both brackets lie to the right of a function
from those that do not. The former I call \emph{clothed}; the latter
are \emph{naked}. Thus the plus sign in $(\sin x+y)$ is naked relative
to the sine (one bracket to the left of the function), but is clothed
in $\sin(x+y)$ (both brackets to the right of the function). 

\subsubsection{Parsing group I}

The only functions in this category are the surd and logical Not. 

Why distinguish the surd from other unary functions? Surely we all
agree that \verb`\sin2\pi`, displaying as $\sin2\pi$, vanishes?
The argument of the sine extends beyond the $2$ to include the $\pi$.
But \verb`\surd2\pi`, displaying as $\surd2\pi$, is understood to
be the product $\surd2\times\pi$. The argument of the surd ends with
the $2$. The surd binds more tightly to its argument than is true
of unary functions generally. 

For parsing group I
\begin{enumerate}
\item if a left bracket is in first position, the mathematical argument
ends with the matching right bracket; otherwise
\item the argument ends with the item in first position and any L- or M-args
required by that item.
\end{enumerate}
If the factorial sign \verb`!` \emph{preceded} its argument, it too
would belong to this parsing state, for it also binds tightly like
the surd. This means that an expression like $\surd4!$ is intrinsically
ambiguous. Is it the square root of $24$ or the factorial of $2$?
In \texttt{numerica} it produces the (perhaps rather odd) error
\begin{centred}
\verb`\eval{$ \surd 4! $}` $\Longrightarrow$ \eval{$ \surd4! $}
\end{centred}
The surd has seized the argument; there is nothing for the factorial
to operate on. The same error arises if the $4$ is parenthesized,
but parenthesizing like either \verb`(\surd 4)!` or \verb`\surd(4!)`
repairs the situation. Because other unary functions (like the sine
or logarithm) do not bind as tightly, this ambiguity does not arise
for them.

Exponents cause no problem because taking square roots and raising
to a power are commutative operations – the result is the same whichever
is performed first. 
\begin{centred}
\verb`\eval{$ \surd 3^4 $}` $\Longrightarrow$ \eval{$ \surd 3^4 $}.
\end{centred}

\subsubsection{Parsing group II: unary functions, slash fractions}

In the default setup this category includes the trigonometric and
hyperbolic functions, their inverses, the various logarithms and the
exponential functions, the signum function \verb`\sgn`, and the slash
fraction \verb`/` where the argument to be determined is its denominator.
Note however that there is a setting switch which enables trigonometric
 functions to handle parentheses in arguments more generally; see
§\ref{subsec:parseTrigFns}. 
\begin{itemize}
\item In parsing group II we wish to accommodate usages like $\ln z^{n}=n\ln z$
(\emph{HMF} 4.1.11), or $\gd z=2\arctan e^{z}-\frac{1}{2}\pi$ (\emph{HMF}
4.3.117), defining the Gudermannian. The exponent is included in the
argument. Considering $\ln(1+1/n)^{n}$ exponents must also be part
of parenthesized arguments. 
\item An approximation to Stirling's formula for the factorial is often
written $\ln N!\approx N\ln N-N$ (widely used in texts on statistical
mechanics). Hence the factorial sign should also be considered part
of the argument.
\item $\ln xy=\ln x+\ln y$ means the argument must reach over a product
of variables. Identities like $\sin2z=2\sin z\cos z$ mean the argument
also reaches over numbers, and expressions like $\sin\tfrac{1}{2}\pi x$
(\emph{HMF} 4.3.104) mean that it further reaches over \verb`\tfrac`-s
and constants.
\item Essentially \emph{anything }can be in first position, and without
parentheses; e.g.
\begin{itemize}
\item unary functions: $\ln\ln z$ (\emph{HMF} 4.1.52), $\ln\tan\dfrac{z}{2}$
(\emph{HMF} 4.3.116),
\item fractions: $\ln\dfrac{z_{1}}{z_{2}}$ (\emph{HMF} 4.1.9), $\arcsin\dfrac{(2ax+b)}{(b^{2}-4ac)^{1/2}}$
(\emph{HMF} 3.3.36), $\ln\dfrac{\tan z}{z}$ (\emph{HMF} 4.3.73),
\item absolute values: $\ln\abs*{\dfrac{a+x}{a-x}}$ (\emph{HMF} 3.3.25),
\item square roots: $\arctan\sqrt{\dfrac{\nu_{1}}{\nu_{2}}F}$ (\emph{HMF
}26.6.8)
\end{itemize}
\end{itemize}
With these examples in mind, for parsing group II
\begin{enumerate}
\item if a left bracket is in first position, the mathematical argument
ends with the matching right bracket and any attached exponent, or
factorial or double factorial sign; otherwise
\item the mathematical argument includes the item in first position and
any L- or M-args required by that item; 
\begin{enumerate}
\item if the item in first position is a number, variable, constant or \verb`\tfrac` 
\begin{enumerate}
\item the argument appends the next item if it is a number, variable, constant
or \verb`\tfrac`, and so on recursively; or
\item the argument appends the next item if it is an exponent, or facorial
or double factorial sign, and ends there; otherwise
\item the argument ends.
\end{enumerate}
\item if the item in first position is not a number, variable, constant
or \verb`\tfrac`
\begin{enumerate}
\item the argument appends the next item if it is an exponent, or factorial
or double factorial sign, and ends there; otherwise
\item the argument ends.
\end{enumerate}
\end{enumerate}
\end{enumerate}
An argument may extend over (see 2(a)i) numbers, constants, variables
and \verb`\tfrac`-s: $\sin2\tfrac{p}{q}\pi x$ exhibits all elements.

Illustrating 1, the exponent is included in the argument but not the
following variable:
\begin{centred}
\verb`\eval{$ \log_{10}(1+2+3+4)^3n $}[n=5]` $\Longrightarrow$ \eval{$ \log_{10}(1+2+3+4)^3n $}[n=5].
\end{centred}
For the sake of the reader, and as one naturally does in any case
to avoid ambiguity, the formula should be written with the variable
$n$ preceding the logarithm: $n\log_{10}(1+2+3+4)^{3}$. The way
the example is written suggests that the writer wished the $n$ to
be considered part of the argument. If that is the case, an outer
set of parentheses would make intentions clear, but it is possible
to leave the argument as written but insert a \verb`\q` command before
$n$: 
\begin{centred}
\verb`\eval{$ \log_{10}(1+2+3+4)^3\q n $}[n=5]` $\Longrightarrow$
\eval{$ \log_{10}(1+2+3+4)^{3}\q n $}[n=5],
\end{centred}
which is $\log_{10}5000.$ 

Illustrating 2(a)ii, again the exponent is included in the argument
but not the following variable:
\begin{centred}
\verb`\eval{$ \log_{10}m^3n $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}m^3n $}[m=10,n=5].
\end{centred}
Again, for the sake of the reader and as one naturally does to avoid
ambiguity, the variable $n$ should precede the logarithm. If in fact
one wants the $n$ included in the argument of the logarithm, the
\verb`\q` command is again available or, better in this case, the
$n$ can be shifted to precede the $m$, which illustrates 2(a)i:
\begin{centred}
\verb`\eval{$ \log_{10}nm^3 $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}nm^3 $}[m=10,n=5].
\end{centred}
Is \texttt{numerica} being too strict when $nm^{3}$ is included in
the argument of the logarithm, but $m^{3}n$ is not? Any criterion
is going to miss some instances where a different outcome might be
desirable. Where an argument ends is affected by visual appearance.
It is simple and easy to remember if it is understood that anything
that breaks the visual appearance of juxtaposed numbers, variables,
constants and \verb`\tfrac`-s ends the argument. An exponent does
just that.

Illustrating 2(b)ii, the argument stops with the \verb`\dfrac` and
its arguments and does not extend to the following constant:
\begin{centred}
\verb`\eval{$ \sin\dfrac12\pi $}` $\Longrightarrow$ \eval{$ \sin\dfrac12\pi $}.
\end{centred}
Obviously, someone writing an expression like this intends the $\pi$
to be part of the argument. In that case, a \verb`\tfrac` should
be used. The \verb`\dfrac` breaks the `visual flow' of an argument. 
\begin{description}
\item [{Fractions}]~

But why not a plain \verb`\frac`? After all, for an inline expression
it displays in the same way as a \verb`\tfrac`. I considered making
the argument-behaviour of \verb`\frac` the same as \verb`\tfrac`
for text-style contexts, and the same as \verb`\dfrac` for display-style
contexts, but that would have meant the same expression evaluating
to different results depending on the context, text-style or display-style,
which ruled it out. Because \verb`\frac` sometimes displays as \verb`\dfrac`,
it necessarily is treated like \verb`\dfrac` (but see §\ref{subsec:parseTrigFns},
specifically \texttt{()=2}). 
\item [{Slash~fractions}]~

It is easy to write ambiguous expressions using the slash $/$ to
indicate fractions or division. How should $\pi/2n$ be interpreted?
With from-the-left evaluation and calculator precedence rules which
give equal precedence to {*} (multiplication) and / (division), this
would be interpreted as $(\pi/2)\times n$, but most people will instinctively
interpret it as $\pi/(2n)$. By placing \verb`/` in parsing group
II, this is what \texttt{numerica} does.

It treats the right-hand argument of the slash \emph{as if it were
the argument of a named function}. This means that $1/2\sin(\pi/6)$
is parsed as $(1/2)\sin(\pi/6)$ rather than as $1/(2\sin(\pi/6))$.
It also means that $1/2\exp(1)$ and $1/2e$ give different results,
which is acceptable since (in the author's view) they display differently
and are not instinctively read in the same way.
\end{description}

\subsubsection{Parsing group III}

\label{subsec:parseTrigFns}By default trigonometric functions are
set to parsing group II. This accommodates many instances of how arguments
are used with these functions, but Fourier series in particular require
more. For them we need to take account of how \emph{parentheses} are
used in arguments. I find $\tan\tfrac{1}{2}(A+B)$ (\emph{HMF }4.3.148),
$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$ (\emph{HMF }19.3.3), $\cos(2m+p)z$
(\emph{HMF }20.2.3), $\sin(2n+1)v$ (\emph{HMF }16.38.1). Looking
through various texts discussing Fourier series it is easy to find
examples like
\[
\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),
\]
and
\[
\cos(N+\tfrac{1}{2})\frac{2\pi\tau}{T},\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
\]
In the last of these \verb`\left` and \verb`\right` have been used
to enlarge the parentheses. 

All these usages can be accommodated by adjusting a setting in the
settings option (§\ref{sec:settingsOption}) of the \verb`\eval`
command:
\begin{lyxcode}
()~=~integer
\end{lyxcode}
where \texttt{integer} is one of \texttt{0, 1, 2}. For convenience
of statement in what follows call parentheses, square brackets or
braces \emph{brackets}. If preceded by a \verb`\left` or \verb`\right`
or \verb`\biggl` or \verb`\biggr` etc. modifier, call them \emph{Brackets},
with an uppercase `B'. Modifiers do not contribute to the position
count, so that a left Bracket in first position means the modifier
and left bracket are both considered to be in first position. When
it is immaterial whether it is a bracket or a Bracket I write b/Bracket.
The rules that follow do not prescribe what mathematicians \emph{ought}
to do but are intended to be descriptive of certain patterns of mathematical
practice as discerned in \emph{HMF} and a number of texts (about half
a dozen) on Fourier series.
\begin{description}
\item [{\texttt{()=0}}] is the \emph{default} setting; b/Brackets are included
in the argument only if
\begin{itemize}
\item the left b/Bracket is in first position;
\begin{itemize}
\item if the first item beyond the matching right b/Bracket is an exponent,
or factorial or double factorial sign, it is appended to the argument,
which ends there, otherwise
\item the argument ends with the right b/Bracket.
\end{itemize}
\end{itemize}
\item [{\texttt{()=1}}] includes a b/Bracketed expression in the argument,
provided 
\begin{itemize}
\item the left Bracket is in first position;
\begin{itemize}
\item if the first item beyond the matching right Bracket is an exponent,
or factorial or double factorial sign, it is appended to the argument,
which ends there, otherwise
\item the argument ends with the right Bracket.
\end{itemize}
\item or the item in first position is a number, variable, constant or \verb`\tfrac`
and the left bracket is in second position;
\begin{itemize}
\item if the first item beyond the matching right bracket is an exponent,
or factorial or double factorial sign, it is appended to the argument,
which ends there, or
\item if the first item beyond the matching right bracket is a number, variable,
constant, or \verb`\tfrac` it is appended to the argument, and so
on recursively, until
\begin{itemize}
\item an exponent, or factorial or double factorial sign is met, which is
appended to the argument which ends there, or
\item an item is met which is \emph{not} an exponent, or factorial or double
factorial sign, or a number, variable, constant or \verb`\tfrac`,
at which point the argument ends, or
\item the end of the formula is reached.
\end{itemize}
\end{itemize}
\end{itemize}
\item [{\texttt{()=2}}] includes a b/Bracketed expression in the argument
provided
\begin{itemize}
\item the left b/Bracket is in first position, or the item in first position
is a number, variable, constant, \verb`\dfrac`, \verb`\frac` or
\verb`\tfrac` and the left b/Bracket is in second position, or the
items in first and second positions are numbers, variables, constants,
\verb`\dfrac`-s, \verb`\frac`-s or \verb`\tfrac`-s and the left
b/Bracket is in third position;
\begin{itemize}
\item if the first item beyond the matching right b/Bracket is an exponent,
or factorial or double factorial sign, it is appended to the argument,
which ends there, or
\item if the first item beyond the matching right b/Bracket is a number,
variable, constant, \verb`\dfrac`, \verb`\frac` or \verb`\tfrac`
it is appended to the argument, and so on recursively, until
\begin{itemize}
\item an exponent, or factorial or double factorial sign is met, which is
appended to the argument which ends there, or
\item an item is met which is \emph{not} an exponent, or factorial or double
factorial sign, or a number, variable, constant, \verb`\dfrac`, \verb`\frac`
or \verb`\tfrac`, at which point the argument ends, or
\item the end of the formula is reached.
\end{itemize}
\end{itemize}
\end{itemize}
\end{description}
The default setting is \texttt{()=0} which is parsing group II behaviour. 

\texttt{()=1} allows $\tan\tfrac{1}{2}(A+B)$ and $\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$,
and $\cos(2m+p)z$ and $\sin(2n+1)v$, and also items on \emph{both}
sides of the bracketed part like $\sin\tfrac{1}{2}(m+n)\pi$ provided
there is only one item between the function and the left bracket:
\begin{centred}
\verb`\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2].
\end{centred}
Note that \texttt{numerica} does not check what is included between
the brackets – it could be anything. However inserting \verb`\left`,
\verb`\right` modifiers before the parentheses restricts the argument
of the sine in this example to the \verb`\tfrac`:
\begin{centred}
\verb`\eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2]`.
$\Longrightarrow$ \eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2].
\end{centred}
\texttt{()=2} draws no distinction between brackets and Brackets.
It allows all \texttt{()=1} possibilities but also \emph{two }items
(of a suitable kind) before the left b/Bracket; it also allows \verb`\dfrac`-s
and \verb`\frac`-s in addition to \verb`\tfrac`-s. 

The following examples are taken from different texts on Fourier series.
The first shows a \verb`\frac` being included in the argument, the
second shows \emph{two} items – including a \verb`\frac` – preceding
the left parenthesis, the third shows a \verb`\frac` to the right
of the parentheses, and the fourth shows parentheses using \verb`\left`-\verb`\right`
modifiers with two items preceding them:
\[
\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),\quad\text{\ensuremath{\sin(N+\tfrac{1}{2})\frac{2\pi\tau}{T}}\ensuremath{\quad}and}\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
\]
All these usages are accommodated by the \verb`()=2` setting. For
instance
\begin{verbatim}
    \eval[p,()=2]
      { 
        \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
      }[N=1,\tau=2,T=3]
\end{verbatim}
$\Longrightarrow$ \eval[p,()=2]
      { 
        \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
      }[N=1,\tau=2,T=3]which is the sine of $(\tfrac{3}{2})\times(\tfrac{4}{3}\pi)=2\pi$
(and \emph{not }$(\sin\tfrac{3}{2})(\tfrac{4}{3}\pi)$ ), where a
\verb`\frac` trailing the parentheses has been included in the argument.
Or consider
\begin{verbatim}
    \eval[p,()=2]
      {\[ 
         \sin2\pi\left(\frac{x}{\lambda}
            -\frac{t}{T}\right) 
      \]}[x=1,\lambda=2,t=3,T=4]
\end{verbatim}
$\Longrightarrow$     \eval[p,()=2]
      {\[ 
         \sin2\pi\left(\frac{x}{\lambda}
            -\frac{t}{T}\right) 
      \]}[x=1,\lambda=2,t=3,T=4] which is the sine of $2\pi\times(-\tfrac{1}{4})=-\tfrac{1}{2}\pi$
(and \emph{not} $\sin2\pi$ times the parenthesised expression) where
there are two items before the parentheses which surround two \verb`\frac`-s
and \verb`\left` and \verb`\right` modifiers have been used with
the parentheses.

However a usage like $\sin(n+\tfrac{1}{2})(x-t)$, noted in two different
texts, is not available without explicit use of the \verb`\q` command
between the parenthesized groups. 

\subsubsection{Parsing group IV}

The only members of this group are \verb`\sum` and \verb`\prod`.

For parsing group IV
\begin{enumerate}
\item the argument ends 
\begin{enumerate}
\item at the first naked plus or minus sign encountered, or 
\item at the first comparison sign or comparison command encountered, or
\item at the first logical And or logical Or sign encountered, or
\item at the end of the formula.
\end{enumerate}
\end{enumerate}
In practice this means mainly (a) and (d), and seems to be the instinctive
practice. \emph{HMF} has multiple examples in multiple chapters of
the argument to a sum ending at a naked plus sign: 7.3.12 \& 7.3.14,
9.1.11 \& 9.1.77, 9.6.35 \& 9.6.43, 11.1.9, \ldots{} (at that point
I stopped looking). They were all of the form
\[
\sum\text{argument}+\ldots
\]
 A minus sign serving the same purpose was harder to find but \emph{HMF}
10.4.65 \& 10.4.67 are two instances. I considered whether a \verb`\times`
or slash fraction sign \verb`/` might end the argument of a sum,
but surely we need to allow things like $\sum1/n^{2}$ which rules
out the slash and \emph{HMF} 9.9.11 provides two of a number of instances
in \emph{HMF} of sum arguments continuing past explicit \verb`\times`
signs (at line breaks when a summand spills onto a second line).

Because they are evaluated using the same code as sums I (unthinkingly)
placed products with sums but doubts later intruded. In \emph{HMF}
products occur only occasionally and are almost all of the form
\[
\prod\left(\text{argument}\right)
\]
where the argument is bracketed (often with \verb`\left \right` modifiers)
and the multiplicand ends with the right bracket. At least twice (\emph{HMF
}6.1.25 and 24.2.2.1) an exponent ($-1$) is attached to the right
bracket and the argument ends there. Looking further afield, a text
on number theory has examples where the argument of the product extends
to \emph{three} parenthesised factors, $\prod\left(\text{arg}1\right)\left(\text{arg2}\right)\left(\text{arg3}\right)$
and a number of others where it extends to two. A text on theory of
functions has 
\[
\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{z/n}
\]
although \emph{HMF}, for the same expression, encloses the two factors
within (large) square brackets, as if some ambiguity existed as to
how far the reach of the \verb`\prod` extended. 

\emph{Tentatively} I retain products here in the same group as sums.

\subsubsection{Parsing group V}

Comparison symbols compose this group: \texttt{=}, \texttt{<}, \texttt{>},
\verb`\ne`, \verb`\le`, \verb`\ge`, \verb`\leq`, \verb`\geq`,
and the various comparison commands from the \texttt{amssymb} package
listed in §\ref{subsec:evalAmssymb-comparisons}. Because of the way
\texttt{numerica} handles comparisons, it is the argument on the right-hand
side of the relation that needs determining.

For parsing group V
\begin{enumerate}
\item the argument ends at
\begin{enumerate}
\item the first logical And or logical Or encountered, or
\item the first comparison sign or command encountered, or
\item the end of the formula.
\end{enumerate}
\end{enumerate}

\subsubsection{Parsing group VI}

Logical And and logical Or are the sole members of this group. It
is the right-hand side of the And or Or command that needs determining.

For parsing group VI
\begin{enumerate}
\item the argument ends at
\begin{enumerate}
\item the first logical And or logical Or encountered, or
\item the end of the formula.
\end{enumerate}
\end{enumerate}

\subsubsection{Disclaimer}

The parsing rules of the different groups are not normative; they
are not statements of how mathematical formulas should be written.
Rather they are attempts to discern regularities in how mathematicians
often do write formulas. It is how things look in the pdf, not \LaTeX ,
that is the guide. You are always free to parenthesize as you see
fit and to insert cleave commands (\verb`\q` or \verb`\Q`) to force
outcomes. 

(But note that parenthesizing has its limits. For sums, writing 
\[
\sum\left(\mathtt{<stuff>}\right)\mathtt{<more-stuff>}
\]
 does not necessarily end the summand at the right parenthesis: it
ends at the first naked $+$ or $-$ sign, or \verb`\Q` command,
encountered.) 

The rule should always be to write expressions that are clear to the
reader of the pdf. An expression that is ambiguous to the reader,
even if it fits within the parsing rules, is to be deplored. The \emph{intent}
is that \verb`\eval` can parse unambiguous expressions correctly. 

\chapter{Nesting \texttt{\textbackslash eval} commands}

\label{chap:Nesting}\verb`\eval` commands can be used within \verb`\eval`
commands, both as part of the formula being evaluated or as part of
the vv-list or both. Only in special circumstances is this likely
to be useful – perhaps when a calculation can be divided into two
or more parts where different settings are appropriate for the different
parts. One can imagine cases in which trigonometric functions are
involved and different \texttt{()} settings would be helpful in different
parts of the formula. Nesting of command within command becomes especially
significant with the additional commands available when \texttt{numerica}
is loaded with the \texttt{plus} or \texttt{tables} options; see the
associated documents \texttt{numerica-plus.pdf} and \texttt{numerica-tables.pdf}.
Since those additional commands are not available in this document,
I restrict myself here to some `toy' examples of \verb`\eval` commands
within \verb`\eval` commands to show how things work. 

\section{Star option for inner \texttt{\textbackslash eval}}

The \verb`\eval` command `digests' a \LaTeX{} formula to produce
an \verb`l3fp`-readable formula. This is then fed to \verb`l3fp`
to be evaluated. The evaluated output is then formatted in various
ways to be displayed. If the inner \verb`\eval` command produces
formatted output, it is \emph{that} that the outer \verb`\eval` command
will attempt to digest – and fail. Hence \emph{always} use the star
option for the inner \verb`\eval` command. That means the outer \verb`\eval`
is feeding on a number only:
\begin{centred}
\verb`\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1 $}` $\Longrightarrow$
\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1$}. 
\end{centred}
Also no math delimiters are used in the inner command. These are irrelevant
with the star option in any case, but in the present context would
cause error if included because they would be treated as part of the
formula and thereby produce an `unknown token' error message. In
the presentation of the overall result that the inner \verb`\eval`
command is evaluated, showing as $0.5$.

\section{Nesting in the vv-list}
\begin{centred}
\verb`\eval{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]` $\Longrightarrow$
\eval{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6].
\end{centred}
When the inner \verb`\eval` command is in the vv-list of the outer
command and has a vv-list of its own, then the entire inner command
needs to be placed in braces:
\begin{centred}
\verb`\eval{$ \sin k\pi + z $}[k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1]`
$\Longrightarrow$ \eval{$ \sin k\pi + z $} [k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1].
\end{centred}
The vv-list of the inner \verb`\eval` command contains both a comma
and square brackets. Both elements need to be hidden from the outer
\verb`\eval` in order that \emph{its }vv-list be parsed correctly.
Hence braces surround the inner \verb`\eval` and its arguments. The
same need arises if the inner \verb`\eval` has a non-empty settings
option – another comma-separated square-bracketed option.

The values of variables used in an inner \verb`\eval` command are
restricted to that command; they do not `leak' into the outer calculation.
But variables and their values in the outer vv-list are available
for use in the inner \verb`\eval` command (unless a value is explicitly
changed in the inner vv-list).

\subsection{Debugging}

\label{subsec:nestDebugging}It is worth looking at the debug display
when \verb`\eval` commands are nested. For the outer \verb`\eval`
command:
\begin{centred}
\verb`\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}` $\Longrightarrow$
\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}
\end{centred}
and when the inner \verb`\eval` is in the vv-list,
\begin{centred}
\verb`\eval[dbg=210]{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]`
$\Longrightarrow$ \eval[dbg=210]{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6]
\end{centred}
For the inner \verb`\eval` command debugging still works but in an
idiosyncratic way. To clarify exactly what is going on I have added
a \verb`\left( \right)` pair around the entire inner \verb`\eval`
command. Note that I have also used a \emph{negative} \texttt{dbg}
value. With a positive value, the right parenthesis is pressed toward
the right margin of the page. The negative value limits the display
to the text width and gives the much neater result shown.
\begin{verbatim}
    \eval[()=2]{$
      \sin\left( 
            \eval*[dbg=-210]{ \sin x }[x=\pi/6]
          \right)\pi + 1 $}
\end{verbatim}
$\Longrightarrow$     \eval[()=2]{$ 
      \sin\left( 
            \eval*[dbg=-210]{ \sin x }[x=\pi/6]
          \right)\pi + 7 $}

\medskip{}
The debug display from the inner \verb`\eval` command has been inserted
into the formula of the outer \verb`\eval` in the position occupied
by the inner \verb`\eval`. I did not deliberately code for this,
but have decided to leave it as is, since there can be no confusion
about which \verb`\eval` command is being `debugged', despite the
potential for some rather odd displays. In this last example, in order
to both use \verb`\left(...\right)` and have the calculation give
the previous result I have employed the setting \verb`()=2` in the
outer \verb`\eval`; see §\ref{subsec:parseTrigFns}.

\chapter{\texttt{\textbackslash nmcInfo (\textbackslash info)}}

\label{sec:info}Used after the evaluation of an `infinite' process,
the \verb`\nmcInfo` command, or its equivalent short-name form \verb`\info`
will tell you how many terms or factors or iterations or steps were
needed to arrive at the result. The syntax of the \verb`\nmcInfo`
command is 
\begin{lyxcode}
\textbackslash nmcInfo\{<arg>\}~
\end{lyxcode}
where \texttt{<arg>} is restricted to one of two choices at present,
either \texttt{sum} or \texttt{prod}. If the package \texttt{numerica-plus.def}
is loaded (see §\ref{subsec:introPackagesOptions} and the associated
document \texttt{numerica-plus.pdf}) two further arguments are possible:
\texttt{iter} and \texttt{solve}. 

There is a starred form of the command:
\begin{lyxcode}
\textbackslash nmcInfo{*}\{<arg>\}
\end{lyxcode}
(or \verb`\info*{<arg>}`). As with the \verb`\eval` command the
star has the effect of suppressing anything other than the numerical
result from the display. 

As an example, let's test a standard identity, $\cosh^{2}x-\sinh^{2}x=1$,
`the hard way'. We know that $\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$
and $\sinh x=x\prod_{k=1}^{\infty}\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)$.
The difference of their squares should be $1$:
\begin{verbatim}
    \eval{\[  
      \left[\sum_{n=0}^{\infty}
        \frac{x^{2n}}{(2n)!}
      \right]^2-
        \left[x\prod_{k=1}^{\infty}
          \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
        \right]^2
      \]}[x=1][3] \info{sum}\quad \info{prod}
\end{verbatim}
$\Longrightarrow$ \eval{\[  
      \left[\sum_{n=0}^{\infty}
        \frac{x^{2n}}{(2n)!}\right]^2-
      \left[x\prod_{k=1}^{\infty}
        \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)\right]^2
    \]}[x=1][3] \info{sum}\quad \info{prod}.

Nearly right. Obviously the product converges only slowly which is
where the error comes from (see the discussion in §\ref{sec:settingsInfiniteSumsProds},
where we needed the extra rounding setting \texttt{P+=3} and $350$
factors to get a correct 3-figure value). The point of the example
is to show the information command being used for both sum and product
in the one evaluation. One does not exclude the other.

The information command can also be placed in the settings option
as the value of the punctuation setting. An example of this has already
been provided earlier which I'll repeat here:
\begin{verbatim}
    \eval[p=\mbox{,\quad\nmcInfo{sum}.}]
      {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3] 
\end{verbatim}
$\Longrightarrow$ \eval[p=\mbox{,\quad\nmcInfo{sum}.}]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]

Because of the \texttt{\textbackslash{[} \textbackslash{]}} delimiters,
if the information command had been placed \emph{after} the \texttt{\textbackslash eval}
command, it would have slid down to the next line. As it is, it resides
\emph{inside} the \texttt{\textbackslash{[} \textbackslash{]}}
delimiters, on the same line as the expression. This may be significant
for adjusting vertical spacing of subsequent elements of the document.

\section{Errors}

Should the wrong argument be used in the \verb`\nmcInfo` command,
no harm is done:
\begin{verbatim}
    \eval{$
      \sum_{k=0}^{\infty}\binom \alpha k x^k
         $}[x=1/2,\alpha=3], \ \info{prod} 
\end{verbatim}
$\Longrightarrow$  \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\ \info{prod}.

$119$ \emph{factors}? The information command is remembering a previous
result, the last time \verb`prod` was used as its argument. Changing
the argument from \verb`prod` to \verb`sum` reveals the correct
number of \emph{terms}.

Should a non-existent argument be used, an error message is generated:
\begin{verbatim}
    \eval{$ 
      \sum_{k=0}^{\infty}\binom \alpha k x^k 
         $}[x=1/2,\alpha=3], \\ \info{Fred} 
\end{verbatim}
$\Longrightarrow$  \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\\  \info{Fred}

\chapter{Saving and reusing results }

\label{chap:Saving-and-retrieving}You may want to use at some place
in a document a result calculated earlier. It would be good to be
able to do so without having to do the calculation again at the new
location. \texttt{numerica} offers a command \verb`\nmcReuse` which
saves a result to a control sequence which can be used elsewhere in
the document, expanding to the saved result. The control sequence
and its content are also saved to file for use on other occasions. 

The syntax of \verb`\nmcReuse` is simple. The command takes two optional
arguments, a star (asterisk) and a conventional square-bracket delimited
argument. If both are used it looks like
\begin{lyxcode}
{\small\textbackslash nmcReuse{*}{[}csname{]}}{\small\par}
\end{lyxcode}
where \texttt{csname} is the proposed name of what will become the
control sequence \verb`\csname` containing the latest result from
the \verb`\eval` command. The name should be composed of letters
only. 

As with \verb`\nmcEvaluate` and \verb`\nmcInfo` there is a short-name
form, \verb`\reuse`, for \verb`\nmcReuse`. If a conflict with another
package arises, it should be possible to fall back on \verb`\nmcReuse`. 

\section{Use without optional argument: \texttt{\textbackslash nmcReuse}}

Suppose your document is \verb`mydoc.tex` (so that the \LaTeX{} command
{\small\verb`\jobname`} expands to \verb`mydoc`). If \verb`\nmcReuse`
is used without optional arguments, then \texttt{numerica} checks
for the existence of a file \verb`mydoc.nmc` in the current document
directory\texttt{ }and if found loads and records the contents of
\verb`mydoc.nmc`. The contents should be a comma separated list of
control sequences and braced values like \texttt{\textbackslash csname1
\{value1\},\textbackslash csname2 \{value2\},...} The control sequences
can then be used elsewhere in the document. In particular, control
sequences containing numerical values can be used in expressions within
\verb`\eval` commands and in vv-lists.\footnote{The associated document \texttt{numerica-plus.pdf} describes how other
quantities like tables and sequences of numerical values (iterates,
recurrences) can also be saved and reused.} 

Note that the control sequences are given \LaTeX{} definitions using
\verb`xparse`'s \verb`\NewDocumentCommand` (a little more general
than \LaTeXe 's \verb`\newcommand`) and can be used at any later
point in the document simply by entering the control sequence (e.g.,
\verb`\csname1`) there (but see §\ref{subsec:reuseGroup-level}).
Should there already be a control sequence with the same name, \LaTeX{}
will generate an error and halt compilation. 

\section{Use with optional name: \texttt{\textbackslash nmcReuse{[}csname{]}}}

If a name \emph{is} supplied, say \verb`\nmcReuse[csname]`, then
not only does \texttt{numerica} first look for \verb`mydoc.nmc` (assuming
your document is called \verb`mydoc.tex`) and load the values stored
in that file if they have not already been loaded, but it also defines
\verb`\csname` to contain the latest result from the \verb`\eval`
command. Should \verb`\csname` already be present in \verb`mydoc.nmc`
and so have been loaded with the other values from \verb`mydoc.nmc`,
the old value is overwritten with the new value using \verb`xparse`'s
\verb`\RenewDocumentCommand` and the new value is saved to the file
\verb`mydoc.nmc`.

\subsection{Group level}

\label{subsec:reuseGroup-level}Control sequences like \verb`\csname`
defined by \verb`\nmcReuse` are defined within the current group
level. A usage like
\begin{centred}
\verb`$ \eval{1+1}\reuse[two] $`
\end{centred}
confines the definition of \verb`\two` to the environment delimited
by the dollar signs. A usage like 
\begin{centred}
\verb`$ \eval{1+1} $ \reuse[two]`,
\end{centred}
where the command has been moved outside the math delimiters, still
confines the definition of \verb`\two` to whatever larger environment
the \verb`\reuse` command might lie within. If in fact the definition
occurs at document level then \verb`\two` (in the present example)
is available for use throughout the document, otherwise it is available
only within the confines of the current environment. 

This is likely to be \emph{not }what is wanted. The remedy is simple:
precede the saved control sequence – \verb`\two` in the present instance
– with a `naked' \verb`\reuse` command. This loads the value stored
in \verb`mydoc.nmc`. In the following example, \verb`\reuse` is
used within a math environment which is followed by some text (\verb`blah blah result:`)
then a `naked' \verb`\reuse` command and the control sequence \verb`\two`
(between math delimiters, which are necessary for reasons explained
in the next section).
\begin{centred}
\verb`$ \eval{1+1} \reuse[two] $, blah blah result: \reuse $\two$.`
$\Longrightarrow$ $ \eval{1+1} \reuse[two]$, blah blah result: \reuse $\two$. 
\end{centred}

\section{What is saved?}

\label{sec:reuseWhat-is-saved}In the default configuration \verb`\nmcReuse`
saves the entirety of the display resulting from the latest \verb`\eval`-uation.
This may include invisible formatting elements meaning that what one
expects to be only a number cannot be inserted into text without causing
a \LaTeX{} error; it requires a math environment to print – see the
last example.

The simplest way to avoid this awkwardness is to use \verb`\eval`
with the star option. This produces a numerical result with \emph{no
}formatting. In the following example, I calculate \verb`11+11` with
\verb`\eval*` and store the value in the control sequence \verb`\twos`.
I then enter some text (\verb`Blah blah blah:`) and insert the control
sequence \verb`\twos` into the text without math delimiters. As you
can see, \verb`\twos` has expanded to $22$, the result of the calculation.
\begin{centred}
\verb`\eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos` $\Longrightarrow$
 \eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos
\end{centred}

\subsection{Viewing what is saved: \textbackslash reuse\texttt{{*}}}

To view what is saved in the \verb`.nmc` file append a star (asterisk)
to the \verb`\reuse` command. (This makes particular sense when using
\texttt{numerica} in a program like \LyX{} with a preview facility;
see Chapter~\ref{chap:LyX}.)
\begin{centred}
\verb`\reuse*` $\Longrightarrow$ \reuse*
\end{centred}
Particularly notable here is the invisible formatting \verb`\mathchoice{}{}{}{}`
accompanying \verb`2` in the value of \verb`\two`. 

The two options of the\verb`\reuse` command can be used together,
in which case the control sequence resulting from the name supplied
in the square-bracketed option will appear in the list resulting from
the star option, i.e. the list contains not just what has been saved
earlier but also the current control sequence saved.

\subsection{\texttt{\textbackslash eval}'s \texttt{reuse} setting}

\label{subsec:reuse=00005Ceval's-reuse-setting}The star option of
the \verb`\eval` command allows a purely numerical result to be saved,
but also only a number is displayed. By using the \verb`reuse` setting
of the \verb`\eval` command it is possible to have both a full display
of an evaluation, vv-list and all, and to save only a numerical result.

For the \emph{starred} form of the \verb`\eval` command it is always
\emph{only the numerical result} that is saved, whatever the value
of the \texttt{reuse} key in the settings option of the \verb`\eval`
command.

For the \emph{unstarred} form of the \verb`\eval` command exactly
what is saved with \verb`\nmcReuse` depends on the \texttt{reuse}
setting:
\begin{lyxcode}
reuse~=~<integer>
\end{lyxcode}
where \verb`<integer>` can take one of two values,
\begin{itemize}
\item \texttt{reuse=0} (the default) saves\emph{ the form that is displayed}
including a formatting component. If the result is displayed in the
form \emph{formula=result (vv-list)} then that is what is saved; if
the display is of the form \emph{result (vv-list)} then that is what
is saved; if the vv-list is empty, an empty formatting component is
still present in the saved result;
\item \texttt{reuse=1} (or, indeed, any non-zero integer) saves only the
numerical result with no other elements of the display (meaning no
formatting component).
\end{itemize}
Thus, with the default setting (\texttt{reuse=0}) the full content
of the display is saved:
\begin{centred}
\verb`\eval{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
\eval{$ x + y $}[x=1,y=1] \reuse[iandi].
\end{centred}
To check that this is the case, \verb`\reuse \iandi` $\Longrightarrow$
\reuse \iandi .

On the other hand, with \texttt{reuse=1} only the numerical value
is saved:
\begin{centred}
\verb`\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi],
\end{centred}
which we can check here: \verb`\reuse\iandi` $\Longrightarrow$ \reuse\iandi . 

\subsubsection{\texttt{\textbackslash reuse} in the preamble}

To gain access from the outset to the control sequences stored in
the file \texttt{mydoc.nmc}, place \verb`\nmcReuse` without an optional
argument in the preamble (but after \verb`\usepackage{numerica}`).

\chapter{Using \texttt{numerica} with \protect\LyX}

\label{chap:LyX}The document processor \LyX{} has a facility that
enables snippets from a larger document to be compiled separately
and the results presented to the user without having to compile the
entire document. The present document was written in \LyX . The demonstration
calculations were evaluated using this \emph{instant preview} facility. 

To use \texttt{numerica} in \LyX{} go to \textsf{Document \lyxarrow{}
Settings \lyxarrow{} LaTeX Preamble} and enter
\begin{lyxcode}
\textbackslash usepackage\{numerica\}
\end{lyxcode}
then click \textsf{OK}. However preview poses problems for the straightforward
use of the \verb`\nmcReuse` command. If you wish to use this command
in \LyX{} then \texttt{numerica} should be loaded with the \texttt{lyx}
package option. Thus in \textsf{Document \lyxarrow{} Settings \lyxarrow{}
LaTeX Preamble} enter
\begin{lyxcode}
\textbackslash usepackage{[}lyx{]}\{numerica\}
\end{lyxcode}
then click \textsf{OK}, or you may wish to follow the above line in
the preamble with \verb`\nmcReuse`,
\begin{lyxcode}
\textbackslash usepackage{[}lyx{]}\{numerica\}

\textbackslash nmcReuse
\end{lyxcode}
and \emph{then} click \textsf{OK}. The additional line ensures all
saved values are available in your document from the outset.

\section{Instant~preview}

Preview performs localised mini-\LaTeX{} runs on selected parts of
a document (for instance, the mathematical parts) and displays the
results in \LyX{} while the user continues to work on the surrounding
document.\texttt{ numerica} uses these local \LaTeX{} runs to do its
evaluations and display their results. That means you get feedback
on your calculations almost immediately.

To use this facility first ensure that instant preview is turned on.
This means selecting \textsf{Tools \lyxarrow Preferences \lyxarrow Look
\& Feel \lyxarrow{} Display} and against \textsf{Instant preview} selecting
\textsf{On}, then clicking \textsf{OK}.

\subsection{Conflict with hyperref support?}

There may be a conflict in \LyX{} between hyperref support and preview,
not that the previews do not form but that their formation takes a
circuitous path that noticeably slows their display.\footnote{At least there is on the author's Windows 10 system, but I'm not sure
that this is general.} If this occurs on your system, go to \textsf{Document \lyxarrow{}
Settings \lyxarrow{} PDF Properties} and ensure the check box \textsf{Use
Hyperref Support} is cleared. By all means reset the check box when
you come finally to compile your document, but until then it should
result in a noticeably brisker display of previews if the checkbox
is cleared.

\section{Mathed}

(Mathed = the \LyX{} mathematics editor.) If you have instant preview
\emph{on} then one way to use \texttt{numerica} in \LyX{} is to enter
an \verb`\eval` command in mathed. Clicking the cursor outside the
editor with the mouse or moving it outside with the arrow keys will
then trigger formation of a preview of the editor's contents – a snippet
of what will be shown in the pdf. This will be displayed in mathed's
place after a generally short `pause for thought' as the mini-\LaTeX{}
run progresses behind the scenes. 

The original expression can be recovered by clicking on the preview.
The content of mathed is immediately displayed and can be edited.

\subsection{\protect\LaTeX{} braces~\{~~\}}

\LyX{} does not support \texttt{numerica}'s \verb`\eval` command `out
of the box' as it does, say, \verb`\frac` or \verb`\sqrt`. To use
the \verb`\eval` command in mathed you will need to supply the braces
used to delimit its mandatory argument. (For \verb`\frac` and \verb`\sqrt`
by contrast, \LyX{} supplies these automatically.) Unfortunately the
\verb`{` key\footnote{\textsf{Shift+{[}} on my keyboard.} does not
insert a left brace into the document but rather an escaped left brace
\verb`\{` as you can see by looking at \textsf{View \lyxarrow{} Code
Preview Pane}. Escaped braces like this are used for grouping terms
in \emph{mathematics}; they are not the delimiters of a \LaTeX{} argument. 

The brace delimiters for \LaTeX{} arguments are entered in mathed by
typing a backslash \textsf{\textbackslash{} }then\textsf{ \{} (two
separate key presses rather than a single combined press). This enters
a balanced pair of (unescaped) braces with the cursor sitting between
them waiting for input. Alternatively, if you have already written
an expression that you want to place between braces, select it, then
type \textsf{\textbackslash{} }then\textsf{ \{}.

\section{Preview insets vs mathed}

There are problems with using mathed for calculations.
\begin{itemize}
\item Expressions entered in mathed are necessarily of the form \verb`$ \eval... $`
or more generally \verb`delimiter` \verb`\eval...` \verb`delimiter`.
But you may wish to wrap the \verb`\eval` command \emph{around} the
math delimiters to produce a \emph{formula=result} form of display.
In mathed the only way to do that is to write the \emph{formula= }part
yourself – which may involve no more than copy and paste but is still
additional mouse work/key pressing. 
\item Mathed does not accept carriage returns. If you want to format a complicated
expression for readability by breaking it into separate lines, you
can't. The expression is jammed into the one line, along with the
settings option content and the vv-list. 
\end{itemize}
For these reasons I have come to prefer \emph{not} using mathed for
calculations but instead to use preview insets wrapped around \TeX -code
(ERT) insets. \LyX{} uses the shortcut \textsf{Ctrl+L} to insert an
ERT inset. Since \LyX{} now does no printing itself, the shortcut \textsf{Ctrl+P}
that was formerly used for printing is available for other purposes.
On my keyboard, the \textsf{P} key lies diagonally up and to the right
but adjacent to the \textsf{L} key. I suggest assigning \textsf{Ctrl+P}
to inserting a preview inset. Then typing \textsf{Ctrl+P Ctrl+L} –
which means holding the \textsf{Ctrl} key down and tapping two adjacent
keys, \textsf{P} followed immediately by \textsf{L} – will insert
an ERT inset inside a preview inset with the cursor sitting inside
the ERT inset waiting for input. In the ERT inset you can enter carriage
returns, and so format complicated expressions. You can place the
vv-list on a separate line or onto consecutive lines. And when you
have finished, clicking outside the preview inset will trigger preview
into doing its thing and present the result `before your eyes'. 

To assign the suggested shortcut, go to \textsf{Tools \lyxarrow{} Preferences
\lyxarrow{} Editing \lyxarrow{} Shortcuts}. Under \textsf{Cursor, Mouse
and Editing Functions} in the main window on the right, scroll down
until you come to \textsf{preview-insert}, select it, then click \textsf{Modify}.
Now press \textsf{Ctrl+P}. The shortcut will magically appear in the
greyed, depressed key.\textsf{ }Click \textsf{OK} and then \textsf{OK}
in the \textsf{Preferences} window to close it. (Most of the examples
in this document have been evaluated in this way, using \textsf{Ctrl+P
Ctrl+L.)}

\section{Errors }

Instant preview will display \texttt{numerica} error messages in \LyX{}
just as it does the results of calculations. Clicking on the message
will show the underlying expression which can then be edited. However
\LaTeX{} errors will \emph{not} produce a preview; formation of the
preview will stall. To find precisely what has gone wrong, you will
need to look at the \LaTeX{} log, but not the log of the overall document;
rather the \emph{preview} log. Unfortunately this is tucked away in
a temporary directory and is not immediately accessible in \LyX{} (unlike
the main \LaTeX{} log from \textsf{Document \lyxarrow{} \LaTeX{} Log}).
When \LyX{} is started, it sets up a temporary directory in which to
perform various tasks. On Windows systems this will be located in
\texttt{C:\textbackslash Users\textbackslash <your name>\textbackslash AppData\textbackslash Local\textbackslash Temp}
and will have a name like \texttt{lyx\_tmpdir.XOsSGhBc1344}. 

One of the tasks \LyX{} uses this temporary directory for is to create
preview images when a document is opened. If you look inside \LyX 's
temporary directory when a document is first loaded, you will see
a subdirectory created, with a name like \texttt{lyx\_tmpbuf0}. There
may already be such directories there, in which case the number on
the end will be greater than \texttt{0} – it depends on whether other
documents are or have been open in the current instance of \LyX .
Inside the appropriate \texttt{lyx\_tmpbuf}\texttt{\emph{n}} folder
will be the preview log with a name like \texttt{lyxpreviewZL1344.log}.
It will usually be accompanied by other files with extensions like
\texttt{.dvi}, \texttt{.tex}, and perhaps quite a number with the
extension \texttt{.png}, each one of which is a preview, or part of
a preview. For a document just loaded there will be only the one preview
log, but if you have added preview insets or math insets to your document\textsf{
}in the current editing session there will be a number of such logs
and you will need to determine the relevant one by the time stamp. 

The log files are text files and can be opened in a text editor. The
relevant part of the log is towards the end (just before the final
statistical summary) where you will find a list of entries like \texttt{Preview: Snippet
1 641947 163840 7864588}. If there is an error, it will be noted here
among these snippets and will generally make clear what needs remedying.

\subsection{CPU usage, \protect\LaTeX{} processes}

It is possible when a preview fails to resolve that the \LaTeX{} process
associated with the preview will continue to run, using CPU cycles,
slowing overall computer performance, and perhaps resulting in extra
fan use giving a different sound to the computer. In Windows 10, the
\textsf{Task Manager} (\textsf{Ctrl+Shift+esc}) under the \textsf{Details}
tab shows the current executables running. The \textsf{CPU} column
will show which processes are preoccupying the CPU. Check whether
one or more of these processes looks \LaTeX -related (e.g. \texttt{latex.exe}
or \texttt{pdflatex.exe}, or \texttt{miktex-pdftex.exe} if using MiK\TeX ).
Click the \textsf{Name} column to sort the processes by name and look
for the relevant name in the list, select it, and end the process
(click the \textsf{End Task} button). 

I am not familiar with the corresponding situation on Linux or Mac.

\section{Using \texttt{\textbackslash nmcReuse}}

As noted, \LyX{} creates its previews in a temporary directory, not
the document directory. If you want to save values from your current
document – say, \texttt{mydoc.lyx} – to \texttt{mydoc.nmc} then you
can do so without drama, but \texttt{mydoc.nmc} will be located in
the temporary directory, and when \LyX{} is closed will be deleted
along with the temporary directory.

Suppose first that at the end of a session you manually copy \texttt{mydoc.nmc}
back to the document directory. How can you ensure that the values
saved in this file are available the next time you open \texttt{mydoc.lyx}?
As noted at the start of this chapter entering 
\begin{lyxcode}
\textbackslash usepackage{[}lyx{]}\{numerica\}

\textbackslash nmcReuse
\end{lyxcode}
in the preamble ensures that these saved values are available for
use from the outset – available to the mini-\LaTeX{} runs creating
previews in the temporary directory.

That leaves the problem of saving new values from the current session,
which are stored in \texttt{mydoc.nmc} in the \emph{temporary} directory,\texttt{
}back to \texttt{mydoc.nmc} in the \emph{document} directory. When
\LyX{} is closed the temporary directory with all its contents is deleted.
As suggested already we could manually copy \texttt{mydoc.nmc} from
the temporary directory to the document directory but that means remembering
to do so before closing \LyX . Inevitably we will sometimes forget. 

Fortunately \LyX{} has a copying mechanism for getting files out of
the temporary directory. When a document is exported – say to pdf
– it is possible to specify a \emph{copier} to automatically copy
back to the document directory or subdirectory various files in the
temporary directory. We want the \texttt{.nmc} file containing the
saved values to be copied back. Go to \textsf{Tools \lyxarrow{} Preferences
\lyxarrow{} File Handling \lyxarrow{} File Formats} and find \textsf{PDF
(pdflatex)} (assuming export to \texttt{pdf} by this route) in the
list of formats. In\textsf{ }the \textsf{Copier} slot of the dialogue
insert the following line of code:
\begin{lyxcode}
{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc,pdf~-d~\$\$i~\$\$o}{\small\par}
\end{lyxcode}
\verb`ext_copy.py` is a python script that is supplied with \LyX .
The \texttt{-e nmc,pdf -d} part of the line tells \texttt{ext\_copy.py}
that on export to \texttt{pdf} by the \texttt{pdflatex} route\texttt{
}to copy any files with the extensions \texttt{.nmc} or \texttt{.pdf}
from the temporary directory where \LyX{} does its work back to the
document directory – the \verb`-d` option (which became available
from \LyX{} 2.3.0). 

But if you have a complex document, it may take too much time to want
to export to pdf before closing \LyX , particularly if there are a
lot of evaluations in the document. Much faster is to export to \emph{plain
text}, not because you want a plain text version of your document
but because it too can be used to trigger the copier mechanism. Go
to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} File Handling \lyxarrow{}
File Formats} and find \textsf{Plain text} in the list of formats.
In the \textsf{Copier} slot enter
\begin{lyxcode}
{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc~-d~\$\$i~\$\$o}{\small\par}
\end{lyxcode}
The only difference from the previous copier command is the absence
of \texttt{pdf}.\footnote{I'm assuming that you don't actually want the plain text version of
the file copied back. If you do, then change \texttt{-e nmc} to \texttt{-e
nmc,txt}.} This will copy \texttt{mydoc.nmc} with its saved values from the
temporary directory back to the document directory. To effect the
export, go to \textsf{File \lyxarrow{} Export }and find \textsf{Plain
text} in the list of formats and click on it. 

A shortcut would be nice. For that go to \textsf{Tools \lyxarrow{}
Preferences \lyxarrow{} Editing \lyxarrow{} Shortcuts}, click on \textsf{New},
enter \texttt{buffer-export text} in the \textsf{Function:} slot,
click on the blank key against \textsf{Shortcut:} and type your shortcut.
You may have to try a number before you find one that hasn't already
been assigned. (I'm using \textsf{Ctrl+}; for no particular reason
beyond the fact that it fits under the fingers easily and saving values
to the document directory has a punctuation-like feel to it, a pause
in the process of writing.) It is now an easy matter to press the
shortcut at the end of a \LyX{} session to copy all the values saved
in \texttt{mydoc.nmc} back to a file of the same name in the document
directory. And it is brisk, not least because plain text export ignores
ERT insets (and hence preview insets wrapped around ERT insets), nor
does it evaluate \verb`\eval` commands in math insets.

\subsection{A final tweak?}

But one still needs to \emph{remember} to press the shortcut. The
thought arises: can \emph{closing} the current document trigger the
copying process? \LyX{} provides a means of linking two commands and
assigning a keyboard shortcut to them with its \texttt{command-sequence}
\LyX{} function. I suggest assigning a shortcut\textsf{ }to
\begin{lyxcode}
command-sequence~buffer-export~text;~view-close
\end{lyxcode}
Indeed, why not reassign the current shortcut for \texttt{view-close},\texttt{
}which is \textsf{Ctrl+W} on my system, to this command sequence?
(I use the \texttt{cua} key bindings – check the \textsf{Bind file:}
slot in \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing \lyxarrow{}
Shortcuts}.)

Please note, however, that \emph{this will work as intended only from
\LyX{} 2.4.0}.\footnote{Due for release in the first half of 2021.}
For \LyX{} 2.3 and earlier, the command sequence will generally fail
because of `asynchronous' processing – \texttt{buffer-export }and
\texttt{view-close} use different threads and the latter may well
start before the former is complete. From \LyX{} 2.4.0 this defect
has been fixed. You press your shortcut, the export to plain text
occurs and the \texttt{.nmc} file is copied back to the document directory,
then the current view is closed.

\section{Using \protect\LyX{} notes}

The central fact about a \LyX{} note is that it does not contribute
to the pdf. But instant preview still works there. This suggests a
possibility: that a calculation be performed within a \LyX{} note and
the result saved using \verb`\nmcReuse` within the same preview inset.
The saved value is now available \emph{from file} for use elsewhere
in the document. In this way, some selected content from a LyX note
\emph{can} find its way into the pdf when the document is compiled.

\chapter{Reference summary}

\section{Commands defined in \texttt{numerica}}
\begin{enumerate}
\item \texttt{\textbackslash nmcEvaluate, \textbackslash eval }
\item \texttt{\textbackslash nmcInfo, \textbackslash info, }
\item \texttt{\textbackslash nmcReuse, \textbackslash reuse}
\item \texttt{\textbackslash q, \textbackslash Q }(`cleave' commands)
\end{enumerate}
Provided they have not already been defined when \texttt{numerica}
is loaded, the following commands are defined in \texttt{numerica}
using \verb`\DeclareMathOperator` from \texttt{amsmath} :
\begin{enumerate}
\item \texttt{\textbackslash arccsc, \textbackslash arcsec, \textbackslash arccot}
\item \texttt{\textbackslash csch, \textbackslash sech}
\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
\textbackslash acsch, \textbackslash asech, \textbackslash acoth}
\item \texttt{\textbackslash sgn, \textbackslash lb}
\end{enumerate}
Provided they have not already been defined, the following commands
are defined in \texttt{numerica} using \verb`\DeclarePairedDelimiter`
from \texttt{mathtools}:
\begin{lyxcode}
\textbackslash abs,~\textbackslash ceil,~\textbackslash floor
\end{lyxcode}
The following commands have been redefined in \texttt{numerica} to
give more spacing around the underlying \verb`\wedge` and \verb`\vee`
symbols:
\begin{lyxcode}
\textbackslash land,~\textbackslash lor
\end{lyxcode}

\section{\textquoteleft Digestible\textquoteright{} content}

\texttt{numerica} knows how to deal with the following content, meaning
that any of these elements occurring within an \verb`\eval` command
should not of itself cause a \texttt{numerica} error. Not all formatting
commands affect display of the output.
\begin{enumerate}
\item variable names (sequences of tokens given values in the variable~=~value
list)
\item digits, decimal point
\begin{enumerate}
\item \texttt{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, .}
\end{enumerate}
\item constants
\begin{enumerate}
\item \texttt{e, \textbackslash pi, \textbackslash gamma, \textbackslash phi,
\textbackslash deg, \textbackslash infty }(sometimes)
\end{enumerate}
\item arithmetic operators
\begin{enumerate}
\item \texttt{+, -, {*}, /, \textasciicircum , \textbackslash times, \textbackslash cdot,
\textbackslash div}
\end{enumerate}
\item logical operators
\begin{enumerate}
\item \texttt{\textbackslash wedge, \textbackslash land, \textbackslash vee,
\textbackslash lor, \textbackslash neg, \textbackslash lnot}
\end{enumerate}
\item comparisons
\begin{enumerate}
\item \texttt{=, <, >, \textbackslash ne, \textbackslash neq, \textbackslash le,
\textbackslash leq, \textbackslash ge, \textbackslash geq}
\item (if \texttt{amssymb} loaded) \texttt{\textbackslash nless, \textbackslash ngtr,
\textbackslash geqq, \textbackslash geqslant, \textbackslash leqq,
\textbackslash leqslant, \textbackslash ngeq, \textbackslash ngeqq,
\textbackslash ngeqslant, \textbackslash nleq, \textbackslash nleqq,
\textbackslash nleqslant}
\end{enumerate}
\item brackets, bracket-like elements, modifiers
\begin{enumerate}
\item \texttt{( ), {[} {]}, \textbackslash\{ \textbackslash\}}
\item \texttt{\textbackslash lparen \textbackslash rparen} (from \texttt{mathtools})\texttt{,
\textbackslash lbrack \textbackslash rbrack, \textbackslash lbrace
\textbackslash rbrace}
\item \texttt{\textbackslash lvert \textbackslash rvert, \textbackslash lfloor
\textbackslash rfloor, \textbackslash lceil \textbackslash rceil}
\item \texttt{| |} (no nesting, deprecated)
\item \texttt{\textbackslash left \textbackslash right, \textbackslash bigl
\textbackslash bigr, \textbackslash Bigl \textbackslash Bigr, \textbackslash biggl
\textbackslash biggr, \textbackslash Biggl \textbackslash Biggr} 
\item \texttt{.} \texttt{/ |} (used with a modifier)
\item \texttt{\textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
\textbackslash ceil{*}\{\}} 
\end{enumerate}
\item unary functions (in the mathematical sense)
\begin{enumerate}
\item \texttt{\textbackslash sin, \textbackslash cos, \textbackslash tan,
\textbackslash csc, \textbackslash sec, \textbackslash cot}
\item \texttt{\textbackslash arcsin, \textbackslash arccos, \textbackslash arctan,
arccsc, \textbackslash arcsec, \textbackslash arccot }
\item \texttt{\textbackslash sin\textasciicircum\{-1\}, \textbackslash cos\textasciicircum\{-1\},
\textbackslash tan\textasciicircum\{-1\}, \textbackslash csc\textasciicircum\{-1\},
\textbackslash sec\textasciicircum\{-1\}, \textbackslash cot\textasciicircum\{-1\}}
\item \texttt{\textbackslash sinh, \textbackslash cosh, \textbackslash tanh,
\textbackslash csch, \textbackslash sech, \textbackslash coth }
\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
\textbackslash csch, \textbackslash sech, \textbackslash acoth}
\item \texttt{\textbackslash sinh\textasciicircum\{-1\}, \textbackslash cosh\textasciicircum\{-1\},
\textbackslash tanh\textasciicircum\{-1\}, \textbackslash csch\textasciicircum\{-1\},
\textbackslash sech\textasciicircum\{-1\}, \textbackslash acoth\textasciicircum\{-1\}}
\item \texttt{\textbackslash exp, \textbackslash lb, \textbackslash lg,
\textbackslash ln, \textbackslash log, \textbackslash log\_\{\},
\textbackslash sgn, \textbackslash surd}
\item \texttt{\textbackslash sqrt\{\}, \textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
\textbackslash ceil{*}\{\}}
\item \texttt{!, !! }(prepended argument)
\end{enumerate}
\item binary functions 
\begin{enumerate}
\item \texttt{\textbackslash tfrac\{\}\{\}, \textbackslash frac\{\}\{\},
\textbackslash dfrac\{\}\{\}}
\item \texttt{\textbackslash tbinom\{\}\{\}, \textbackslash binom\{\}\{\},
\textbackslash dbinom\{\}\{\}}
\item \texttt{\textbackslash sqrt{[}{]}\{\}}
\end{enumerate}
\item $n$-ary functions
\begin{enumerate}
\item \texttt{\textbackslash min, \textbackslash max, \textbackslash gcd}
\end{enumerate}
\item sum, prod
\begin{enumerate}
\item \texttt{\textbackslash sum\_\{\}\textasciicircum , \textbackslash prod\_\{\}\textasciicircum{} }
\end{enumerate}
\item formatting commands
\begin{enumerate}
\item \texttt{,} (comma, in $n$-ary functions)
\item \texttt{\{\}, \textbackslash\textbackslash , \&, \textbackslash to}
\item \texttt{\textbackslash dots, \textbackslash ldots, \textbackslash cdots,}
\item \texttt{\textbackslash{} , \textbackslash ,{}, \textbackslash ;,
\textbackslash :, \textbackslash !, \textbackslash >}
\item \texttt{\textbackslash thinspace, \textbackslash quad, \textbackslash qquad
, \textbackslash hfill, \textbackslash hfil}
\item \texttt{\textbackslash phantom\{\}, \textbackslash vphantom\{\},
\textbackslash hphantom\{\}}
\item \texttt{\textbackslash xmathstrut{[}{]}\{\}} \texttt{, \textbackslash splitfrac\{\}\{\},
\textbackslash splitdfrac\{\}\{\} }(from \texttt{mathtools}), \texttt{\textbackslash mathstrut}
\item \texttt{\textbackslash displaystyle, \textbackslash textstyle, \textbackslash scriptstyle,
\textbackslash scriptscriptstyle}
\item \texttt{\textbackslash label\{\}, \textbackslash ensuremath\{\},
\textbackslash text\{\}, \textbackslash mbox\{\}}
\item \texttt{\textbackslash begin\{\}, \textbackslash end\{\}}
\end{enumerate}
\item font commands
\begin{enumerate}
\item \texttt{\textbackslash mathrm\{\}, \textbackslash mathit\{\}, \textbackslash mathcal\{\},
\textbackslash mathtt\{\}, \textbackslash mathbf\{\}, \textbackslash mathbb\{\},
\textbackslash mathsf\{\}, \textbackslash mathfrak\{\}, \textbackslash mathscr\{\},
\textbackslash mathnormal\{\}, \textbackslash boldsymbol\{\}}
\end{enumerate}
\end{enumerate}

\section{Settings}

\subsection{Available \texttt{\textbackslash nmcEvaluate} settings}
\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
\midrule
{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
{\small\texttt{o}} &  & {\small degree switch for trig. funcions} & \tabularnewline
{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
{*} &  & {\small switch to suppress equation numbering (if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
in }{\small\texttt{vvd}}{\small )} & \tabularnewline
{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion, sums} & {\small\texttt{2}}\tabularnewline
{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, sums} & {\small\texttt{0}}\tabularnewline
{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion, products} & {\small\texttt{2}}\tabularnewline
{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, products} & {\small\texttt{0}}\tabularnewline
{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}

\subsection{Available configuration file settings}
\begin{center}
\bigskip{}
\begin{tabular}{ll}
\toprule 
key & default\tabularnewline
\midrule
{\small\texttt{rounding}} & {\small\texttt{6}}\tabularnewline
{\small\texttt{pad}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{output-sci-notation}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{output-exponent-char}} & {\small\texttt{e}}\tabularnewline
{\small\texttt{input-sci-notation}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{input-exponent-char}} & {\small\texttt{e}}\tabularnewline
{\small\texttt{multitoken-variables}} & {\small\texttt{1}}\tabularnewline
{\small\texttt{logarithm-base}} & {\small\texttt{10}}\tabularnewline
{\small\texttt{intify-rounding}} & {\small\texttt{14}}\tabularnewline
{\small\texttt{vv-display}} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
{\small\texttt{vv-inline}} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
{\small\texttt{sum-extra-rounding}} & {\small\texttt{2}}\tabularnewline
{\small\texttt{sum-query-terms}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{prod-extra-rounding}} & {\small\texttt{2}}\tabularnewline
{\small\texttt{prod-query-terms}} & {\small\texttt{0}}\tabularnewline
{\small\texttt{eval-reuse}} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{document}
